point-set and point-free topology - MIMS - The University of

POINT-SET AND POINT-FREE

TOPOLOGY IN CONSTRUCTIVE SET

THEORY

A thesis submitted to the University of Manchester

for the degree of Doctor of Philosophy

in the Faculty of Engineering and Physical Sciences

2005

Christopher M Fox

School of Mathematics

Contents

Abstract 5

Declaration 6

Copyright 7

Acknowledgements 8

1 Introduction 9

1.1 Intuitionism, constructivism and CZF . . . . . . . . . . . . . . . . . . 9

1.2 Point-free topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Constructive Topological Spaces 28

2.1 Basic pairs and ct-spaces . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Constructive separation axioms . . . . . . . . . . . . . . . . . . . . . 33

2.3 Sobriety and weak sobriety . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Formal Topology 40

3.1 Formal topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Three Galois adjunctions in CZF . . . . . . . . . . . . . . . . . . . . 43

3.3 Three equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Quasi-formal topologies . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.6 Separation properties and compactness . . . . . . . . . . . . . . . . . 83

2

4 Inductively Generated Formal Topologies 88

4.1 Covering systems and set-presentability . . . . . . . . . . . . . . . . . 88

4.2 Constructions on inductively generated formal topologies . . . . . . . 101

4.3 Application to a result of Coquand . . . . . . . . . . . . . . . . . . . 111

4.4 Generalisation to quasi-formal topologies . . . . . . . . . . . . . . . . 118

4.5 Coinductively generating a binary positivity predicate . . . . . . . . . 121

5 Metric Spaces and Metric Formal Topologies 128

5.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2 Metric formal topologies . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.3 A Galois adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.4 Metric formal topology completions . . . . . . . . . . . . . . . . . . . 149

5.5 Compactness and total boundedness . . . . . . . . . . . . . . . . . . 157

5.6 Metric subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6 Uniform Spaces and Uniform Formal Topologies 168

6.1 Uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.2 Uniform formal topologies . . . . . . . . . . . . . . . . . . . . . . . . 181

6.3 Two Galois Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.4 Gauges and metrisation . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.5 Uniform formal topology completions . . . . . . . . . . . . . . . . . . 199

6.6 Compactness and total boundedness . . . . . . . . . . . . . . . . . . 210

7 Balanced Formal Topologies and the Basic Picture 213

7.1 Introduction to the Basic Picture . . . . . . . . . . . . . . . . . . . . 213

7.2 Problems with finding “enough” axioms . . . . . . . . . . . . . . . . 216

7.3 The category FCFrm in IZF . . . . . . . . . . . . . . . . . . . . . . 220

7.4 Continuous relations between basic pairs . . . . . . . . . . . . . . . . 225

8 Further Work 232

3

A The axioms of CZF and its extensions 234

A.1 Class notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

A.2 The axioms of CZF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

A.3 Choice principles and Relation Reflection . . . . . . . . . . . . . . . . 236

A.4 Extension axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Bibliography 240

Index 248

Word count of Chapters 1–8 and Appendix A1,

excluding mathematical formulae: 52,923

including mathematical formulae: 77,728

1Based on the output from untex -i -m and untex -i. The true word count is likely to besomewhere between these two figures.

4

Abstract

Since the 1970s, considerable progress has been made in point-free topology using

locale theory, in both the classical and the impredicative constructive context. In

contrast, the predicative approach to point-free topology, known as formal topology,

remains some way behind impredicative locale theory.

One of the principal aims of this thesis is to show how some of the key ideas

and results from locale theory can be transferred to formal topology, working in

Constructive Zermelo-Fraenkel Set Theory (CZF). Building on the work of [7], we

also examine, in the constructive predicative case, the relationship between various

structures in point-set and point-free topology.

In Chapter 6 we study uniform formal topologies; as well as making a contribution

to the predicative theory, some of this work goes beyond what was previously known

about uniform locales in the impredicative constructive case.

Chapter 7 explores the relatively new field of the Basic Picture, including the study

of formal topologies with a binary positivity predicate. We prove some results which

shed some light on the similarities and differences between this and the traditional

approach to point-free topology.

5

Declaration

No portion of the work referred to in this thesis has been

submitted in support of an application for another degree

or qualification of this or any other university or other

institution of learning.

6

Copyright

Copyright in text of this thesis rests with the Author. Copies (by any process)

either in full, or of extracts, may be made only in accordance with instructions given

by the Author and lodged in the John Rylands University Library of Manchester.

Details may be obtained from the Librarian. This page must form part of any such

copies made. Further copies (by any process) of copies made in accordance with such

instructions may not be made without the permission (in writing) of the Author.

The ownership of any intellectual property rights which may be described in this

thesis is vested in the University of Manchester, subject to any prior agreement to

the contrary, and may not be made available for use by third parties without the

written permission of the University, which will prescribe the terms and conditions

of any such agreement.

Further information on the conditions under which disclosures and exploitation

may take place is available from the Head of the School of Mathematics.

7

Acknowledgements

Firstly I would like to thank my supervisor, Peter Aczel, for his guidance over the

last three years, and the Engineering and Physical Sciences Research Council for their

generous financial support; without them this thesis would not have been possible. I

am also grateful to the School of Mathematics for their support, and to the Math-

ematical Foundations Group in the School of Computer Science for allowing me to

participate in their seminars.

I would like to express my thanks to the organisers of the Workshop on Philosophy

of Mathematics and the Symposium on Constructive Mathematics, held at Uppsala

University in 2004, and to all the speakers at both events. By attending the conference

I gained a better understanding of several subjects relevant to my research.

I also thank everyone in the University of Manchester Hiking Club for helping

me to relax, my friends from Cambridge for staying in touch, and finally my mother,

Christine, and my late father, Martyn, for their encouragement.

8

Chapter 1

Introduction

1.1 Intuitionism, constructivism and CZF

1.1.1 Bishop’s constructivism

There is no single set of beliefs which can be labelled as “constructivism”. By the

mid-20th century, several schools of constructivism had emerged, each with its own

beliefs which contradicted those of the other schools. In some cases principles were

admitted which were not valid in classical mathematics, and it is likely that this acted

as a barrier to a more widespread adoption of constructive mathematics.

Furthermore, one common characteristic of many varieties of constructivism was

the rejection of the Principle of Excluded Middle (PEM):

φ ∨ ¬φ

where φ is any formula. It was believed by most classical mathematicians at the time

that it was impossible to carry out any serious mathematics without PEM.

Beginning in the 1960s with the publication of Foundations of Constructive Anal-

ysis [14], Errett Bishop addressed these problems by adopting a set of principles

which were valid not only in classical mathematics, but also in L. E. J. Brouwer’s

Intuitionism and in Constructive Recursive Mathematics. Bishop was able to con-

structivise several branches of analysis in this framework. A second edition of his

9

CHAPTER 1. INTRODUCTION 10

book, entitled Constructive Analysis [15], was completed by Douglas Bridges in 1985

shortly after Bishop’s death.

Before describing Bishop’s constructivism, we shall summarise some of the impor-

tant points concerning Intuitionism and Constructive Recursive Mathematics. For

more details, two good starting points are [82] and [83].

Brouwer’s Intuitionism

Brouwer’s main beliefs are summarised by Troelstra in [82] as follows:

• “Mathematics is not formal; the objects of mathematics are mental construc-

tions in the mind of the (ideal) mathematician. Only the thought constructions

of the (idealized) mathematician are exact.”

• “Mathematics is independent of experience in the outside world, and mathemat-

ics is in principle also independent of language. Communication by language

may serve to suggest similar thought constructions to others, but there is no

guarantee that these other constructions are the same.”

• “Mathematics does not depend on logic; on the contrary, logic is part of math-

ematics.”

Brouwer’s Intuitionism accepts the Brouwer-Heyting-Kolmogorov (BHK) inter-

pretation of the logical operators. In particular, to prove A∨B one must give a proof

of A or a proof of B, and specify which of the two propositions has been proved. To

prove A → B, one must give a construction which transforms a proof of A into a

proof of B. Under this interpretation, if PEM were true then we would be able to

determine whether any given formula is true or false.

One distinctive feature of intuitionism is the acceptance of choice sequences. These

are sequences of natural numbers which may be either lawlike (determined by some

rule), or lawless (chosen arbitrarily by the creative subject). The use of choice se-

quences, combined with the intuitionistic interpretation of logic, led Brouwer to ac-

cept the continuity axiom, which asserts that if f is a natural number-valued function


defined on a collection of choice sequences, then the value of f at any choice sequence

depends on a finite initial segment of the sequence. This is an example of an intu-

itionistic principle which is not valid classically.

Another principle which is heavily used in Brouwer’s Intuitionism is the Fan The-

orem, which can be proved from Brouwer’s Bar Induction principle or taken as an

axiom in its own right. The Fan Theorem is equivalent to the statement that the

Cantor space 2N is compact. More explicitly, it says that if T is a finitely branching

tree in which every node has at least one successor, A ⊆ T and every infinite branch

of the tree passes through A, then there is an n ∈ N such that every branch passes

through A in at most n steps. Although the Fan Theorem is compatible with classical

logic, it is not valid in Constructive Recursive Mathematics.

A detailed discussion of the Bar and Fan Theorems can be found in [83].

Constructive Recursive Mathematics

Constructive Recursive Mathematics (CRM) is a more concrete form of construc-

tivism in which the objects of mathematics are words in various alphabets. Proofs in

CRM are based around a precise notion of an algorithm. In order to prove A → B,

we must give an algorithm which, given a proof of A, will produce a proof of B. A

proof of (∀x)A(x) is an algorithm which, given an object x, will produce a proof of

A(x). To prove (∃x)A(x) we must provide a object x and a proof of A(x).

Two principles which are accepted in CRM are Church’s Thesis (CT) and Markov’s

Principle (MP). Church’s Thesis is the principle that every total function f : N → N

is recursive, that is, there is an m ∈ N such that the mth recursive function fm is

total and fm(n) = f(n) for all n. Markov’s Principle asserts that if a computation

does not run for ever then then it must terminate after some finite number of steps;

equivalently if f : N → 0, 1 is recursive then

¬(∀n ∈ N)[f(n) = 0] → (∃n ∈ N)[f(n) = 1]

Church’s Thesis is clearly not valid in classical mathematics. Furthermore it can be

shown that CT is incompatible with Brouwer’s Fan Theorem, because it is possible to


construct a recursive binary tree with arbitrarily long finite branches but no infinite

recursive branch. This was first proved by Kleene in 1952, and a constructive proof

can be found in [13]. Markov’s Principle is valid classically, but is not valid in

Brouwer’s intuitionism.

Bishop’s Constructivism

As mentioned earlier, the theorems of Bishop’s constructive mathematics (BCM) are

valid when interpreted in Brouwer’s Intuitionism, Constructive Recursive Mathemat-

ics or classical mathematics. Bishop was able to avoid the use of principles such as

Bar Induction and Church’s Thesis by choosing his definitions carefully whenever

there was a choice between several classically equivalent alternatives. For example,

Brouwer used the Fan Theorem to prove that a continuous function f : R → R is uni-

formly continuous on every closed bounded interval. Bishop avoided this by defining

a function to be continuous if and only if it is uniformly continuous on every closed

bounded interval.

The underlying principles of Bishop’s constructivism are set out in Chapter 1 of

[15]. Of particular interest to us is the notion of a set. In order to define a set,

we must describe firstly what must be done to construct an element of the set, and

secondly what must be done to prove that two elements of the set are equal. To

construct a function from a set A to a set B, we must prescribe a “finite routine”

which, given an element of A, produces an element of B, and prove that equality is

preserved.

It is noted in [15] that the axiom of choice can apparently be justified by the

constructive interpretation of the logical operators. If (∀x ∈ A)(∃y ∈ B)φ(x, y) then

by the meanings of the quantifiers ∀ and ∃ there must be an operation which, given

an object x ∈ A, produces a y ∈ B and a proof of φ(x, y). However, there is no

guarantee that this operation will preserve equality: two equal elements of A may

give rise to elements of B which are not equal.


Foundations for Bishop-style mathematics

Bishop did not explicitly describe a formal system on which his version of construc-

tivism should be based. Two possible foundations for BCM were put forward in the

first half of the 1970s.

The first of these was Martin-Lof’s Intuitionistic Type Theory (ITT), described

in [56] and [57], which is still in use by many constructivists (including formal topol-

ogists) today. This is a dependent type theory which includes basic types such as

the natural numbers, together with dependent types such as type-indexed products

(Πx : A)B(x) and sums (Σx : A)B(x).

One central idea in ITT is that of propositions-as-types. A type A can be viewed

as the proposition that A is inhabited, and an element x : A can be viewed as a proof

of this proposition. Conversely, a proposition A corresponds to the type of proofs of

A. Under this correspondence, the propositions A & B and A∨B correspond to the

types A × B and A + B. Implication A → B corresponds to the type of functions

from A to B. Universal quantification (∀x : A)B(x) corresponds to the product type

(Πx : A)B(x), and existential quantification (∃x : A)B(x) corresponds to the sum

type (Σx : A)B(x).

A consequence of this is that the Axiom of Choice is a theorem of ITT, because

a proof of (∀x : A)(∃y : B)φ(x, y) is essentially the same as a function which takes

an element x : A to a pair consisting of an element y : B and a proof of φ(x, y).

As well as the types mentioned above, it is common to include a type U of small

types. Any type which can be constructed without referring to U is considered small.

In the original version of ITT, a type of all types was included, but this was found to

be inconsistent with the propositions-as-types principle due to a version of Russell’s

Paradox.

ITT is a good foundation for constructive mathematics because of the propositions-

as-types principle, which makes the BHK interpretation very explicit. However, the

process of formalising mathematics in ITT is considerably different to that of working


in Zermelo-Fraenkel set theory, making ITT less accessible to classical mathemati-

cians. Simple operations such as defining a subset by separation require much more

work in ITT, although some progress has been made by Sambin and Valentini in [79],

and by Carlstrom in [19] and [20].

Another early foundation for BCM is Myhill’s Constructive Set Theory (CST) [59].

This is an extensional set theory with three distinct sorts: natural numbers, sets and

functions. CST is more similar in style to ZF set theory, and allows many standard

set-theoretical constructions (e.g. unions, intersections, products) to be carried out

with little difficulty. While ZF is a theory in a first-order language with one two-place

predicate ∈, CST is far more complicated, with three constants, four predicates and

22 axioms and axiom-schemes (excluding the two optional choice axioms).

A common characteristic of ITT and CST is that impredicative definitions of

sets are forbidden. Although Bishop makes no explicit mention of impredicativity

in [14, 15], it was felt by some authors, including Myhill [59], that Bishop’s notion

of a set ruled out definitions of sets which involve quantification over the class of

all sets, because such definitions do not unambiguously specify what must be done

in order to construct an element of the set in question. This rules out the use of

the Powerset axiom, along with the full Separation axiom-scheme. Rather than omit

the Separation axiom-scheme altogether, it is weakened to the Restricted Separation

axiom-scheme (also called ∆0-Separation or Predicative Separation). A formula is

restricted if it only contains quantifiers which are restricted to a set, i.e. those of the

form (∀x ∈ a) and (∃x ∈ a). Restricted Separation asserts that if a is a set and φ(x)

is a restricted formula, then x ∈ a |φ(x) is a set.

1.1.2 Constructive Set Theory

Another foundation for Bishop’s constructive mathematics is Aczel’s Constructive

Zermelo-Fraenkel Set Theory (CZF). CZF is another extensional set theory which has

some similarities with Myhill’s CST. Like classical ZF, CZF has just one sort (that

of sets), and a two-place predicate ∈. The axioms of CZF are listed in Appendix A;


the differences from ZF are listed below:

• Classical logic is replaced by Intuitionistic Logic

• The axiom of Foundation is replaced by the Set Induction scheme

• The axiom-scheme of Separation is weakened to that of Restricted Separation

• The Powerset axiom is omitted

• The axiom-schemes of Strong Collection and Subset Collection are introduced

Let IZF be the theory obtained by adding the Powerset and full Separation axioms

to CZF. In the presence of classical logic, CZF, IZF and ZF are equivalent.

CZF has an interpretation in a form of Martin-Lof type theory to which a type

of sets has been added. The details of this interpretation can be found in [1, 2, 3].

A detailed description of how to carry out many of the basic constructions of math-

ematics in CZF, including ordered pairs, cartesian products of sets, exponentiation

and the construction of the natural, rational and real numbers, is given in [7].

Class notation is used throughout [7] and this thesis to make the mathematics

more readable. The main points of class notation are covered in Appendix A. Because

of the absence of the Powerset and full Separation axioms, many structures that we

wish to consider will be proper classes (that is, it cannot be proved that they are

sets); for example, the collection of open sets of any inhabited topological space will

always be a proper class in CZF. Rather than introduce terminology such as “class-

relation”, “class-function” and “class-frame”, we shall capitalise the first letter of

a word to indicate that an object may be a class rather than a set, for example,

“Relation”, “Function” and “Frame”1.

Non-constructive principles and the axiom of choice

Methods for showing that certain principles are not valid in CZF include sheaf models

[38] and realisability [58, 70]. Where possible, if we wish to show that a proposition

1Thanks are due to Paul Taylor for suggesting this notation.


is not constructively valid, we will attempt to show that implies the truth of some

well-known non-constructive principle such as those listed below, all of which can be

disproved using either of the above methods.

• The Principle of Excluded Middle (PEM) asserts that φ ∨ ¬φ for any formula

φ.

• The Restricted Principle of Excluded Middle (REM) asserts that φ∨¬φ for any

restricted formula φ; equivalently any subset of 0 is equal to either ∅ or 0.

• The Weak Restricted Principle of Excluded Middle (wREM) asserts that ¬φ ∨

¬¬φ for any restricted formula φ.

• The Limited Principle of Omniscience (LPO) asserts that (∀n ∈ N)[f(n) =

0] ∨ (∃n ∈ N)[f(n) = 1] for every function f : N → 0, 1.

Given a set X, there is a bijection between Pow(X) and the class of functions

from X to Pow(0). It follows that if Pow(0) is a set then Pow(X) is a set

for every set X. Thus if some principle implies that Pow(0) is a set, then that

principle implies the Powerset axiom and is therefore not valid in CZF.

Some choice principles, including Countable Choice and Dependent Choice (see

Appendix A), are valid both in the type-theoretic interpretation of CZF and in re-

alisability models. However, we prefer to avoid all choice principles where possible

because they are not valid in topos mathematics, nor do they hold in sheaf models of

CZF. The full Axiom of Choice implies REM (see [9]) and can therefore never be used

constructively. Many proofs in ITT use a technique involving type-theoretic choice

to construct sets without using the Powerset axiom. In CZF, we can often use the

Strong Collection and Subset Collection schemes instead.

It was shown in [58] and [70] that there are realisability models of IZF and CZF

respectively in which Church’s Thesis is valid. We can therefore show that a principle

is not provable in CZF or IZF by showing that it is inconsistent with CT; an example

of this can be found in Proposition 7.4.4. Realisability models can also be used to

validate MP, provided classical logic is assumed in the background theory.


Inductive definitions

We will often need to define sets or classes inductively. An inductive definition is a

class of ordered pairs. If Φ is an inductive definition, a class Y is Φ-closed if

(∀X)(∀a)[〈X, a〉 ∈ Φ & X ⊆ Y → a ∈ Y ]

The problem of inductive definitions is to show that there is a smallest Φ-closed class,

and that under certain conditions this class is a set. This is discussed in [3] and [9],

and the main results from [9] are stated below.

To inductively define sets (rather than classes), we will need the Regular Extension

Axiom (REA), or the Weak Regular Extension Axiom (wREA), which are explained

in Appendix A. REA and wREA are valid in the type-theoretic interpretation of

CZF, provided the underlying type theory has W -types, which allow types to be

constructed using a form of inductive definition (see [3]).

Theorem 1.1.1 For any inductive definition Φ there is a smallest Φ-closed class

I(Φ).

Definition 1.1.2 A class B is a bound for an inductive definition Φ if whenever

〈X, a〉 ∈ Φ there is a set b ∈ B and a surjection from b onto X.

Φ is bounded if it has a bound that is a set, and a | 〈X, a〉 ∈ Φ is a set for each

set X.

Theorem 1.1.3 (CZF + wREA) If Φ is a bounded inductive definition then the

smallest Φ-closed class I(Φ) is a set.

If Φ is an inductive definition and X is a class, we can form a new inductive

definition

ΦX = Φ ∪ 〈∅, a〉 | a ∈ X

Clearly Y is ΦX-closed if and only if X ⊆ Y and Y is Φ-closed. Let I(Φ, X) = I(ΦX),

the smallest Φ-closed class containing X. If Φ is bounded and X is a set, then ΦX

is bounded, so I(Φ, X) is a set in the presence of wREA. Furthermore, if REA is

assumed, we have the following.


Theorem 1.1.4 (CZF + REA) If S is a set and Φ is an inductive definition that

is a subset of Pow(S) × S, then there is a set B of subsets of S such that for each

a ∈ S and each subclass X ⊆ S,

a ∈ I(Φ, X) ⇒ (∃Y ∈ B)[Y ⊆ X & a ∈ I(Φ, Y )]

1.2 Point-free topology

1.2.1 Point-free topology and locale theory

One of the earliest uses of point-free methods in constructive mathematics was in

[55]. For various topological spaces, such as the real numbers, the Cantor space 2N

and the Baire space NN, open sets were represented by collections of basic open sets

(open intervals in R with rational endpoints, or sets of all sequences in 2N or NN

beginning with a certain prefix), and the notion of one open set covering another

was defined in terms of these collections of basic open sets, without referring to the

points of the space. Classically this notion of covering coincided with the ordinary

point-set notion, but the two notions differed constructively. The point-free notion

allowed constructive proofs of results such as the Heine-Borel theorem, and a version

of Brouwer’s Fan Theorem valid in CRM.

A more consistent approach to point-free topology is that of Locale Theory, which

is now the most widely used form of point-free mathematics. One of the first papers

on Locale Theory was [42], and one of the standard reference books on the subject

is [45]; the reader may also find [88] helpful.

Frames and locales

A frame is a poset A with a top element (>), binary meets (∧) and joins of all subsets

of A (∨

), satisfying the infinite distributive law:

a ∧∨

S =∨a ∧ b | b ∈ S


for all a ∈ A and S ⊆ A. If A and B are frames, a frame homomorphism A→ B is a

function from A to B preserving the structure of the frame (≤,>,∧,∨

). The frames

and frame homomorphisms form a category Frm.

The elements of a frame can be thought of as open sets in an “imaginary” topo-

logical space, with a ≤ b if and only if a is contained in b. If X is a topological space

then the lattice of open sets ΩX is a frame. Given a continuous function f : X → Y ,

the map V 7→ f−(V ) is a frame homomorphism ΩY → ΩX. This gives rise to a

functor Ω : Top → Frmop. We let Loc = Frmop, the category of locales.

Conversely, starting with a locale A, we can construct a topological space Pt(A),

whose points are the locale morphisms 1 → A (that is, the frame homomorphisms

A → 1), where 1 = Pow(0) ordered by inclusion. The open sets of the topology

are those of the form p ∈ Pt(A) | 0 ∈ p(a) for each a ∈ A. Pt can be extended to

a functor Pt : Loc → Top by setting Pt(f)(p) = p f when Bf→ A

p→ 1 in Frm.

One of the fundamental results of Locale Theory is the existence of an adjunction

Ω a Pt. Given a topological space X, the space Pt(ΩX) will not necessarily be

isomorphic to X. If the unit ηX : X → Pt(ΩX) is a homeomorphism, we say that

the space is sober. Every sober space is T0, and classically every T2 space is sober,

but constructively the space of rational numbers with the Euclidean topology fails

to be sober. The adjunction has the property that Pt(A) is sober for every locale A,

and the functors Ω and Pt restrict to an equivalence between the full subcategories

of Top and Loc whose objects are the sober spaces and spatial locales (locales of

the form ΩX) respectively. Constructive sobriety will be discussed in more detail in

Sections 2.3 and 3.2.

Much of the early interest in locales was concerned with finding classical proofs

which avoided the axiom of choice. A well-known example is the Tychonoff theorem,

which states that a product of compact topological spaces is compact. The proof for

topological spaces relies on the axiom of choice, but if locales are used instead (with

a suitable point-free notion of compactness) then a point-free version of the theorem

can be proved without the use of choice. It was also noticed that locale-theoretic

proofs are often valid in intuitionistic mathematics, even when the corresponding


point-sensitive proofs require classical logic.

1.2.2 Predicative formal topology

Although proofs in Locale Theory are less likely to require classical logic than their

point-sensitive counterparts, Locale Theory has the drawback that it is impredicative,

and therefore cannot be used in ITT or CZF without some modification. In the

absence of the Powerset axiom, all but the most trivial locales fail to be a set.

A solution to this problem is to use formal topologies in place of locales. These

were first introduced by Sambin in [73], based on the notion of a formal space used

by Fourman and Grayson (without any explicit mention of predicativity) in [36]. The

definition of a formal topology was later refined in [72], and adapted from ITT to

CZF in [7].

A formal topology is a set S, together with a Relation / ⊆ S × Pow(S) and, in

some definitions, a preorder ≤ on S, satisfying some axioms listed in Chapter 3. The

intention is that S represents a base of a certain frame, that is, every element of the

frame can be expressed as a join of some elements of S. For a ∈ S and U ∈ Pow(S),

a / U means that a is covered by the join of U .

A formal topology may also have a unary positivity predicate, Pos ⊆ S, which

expresses that an open set is inhabited, or a binary positivity predicate n ⊆ S ×

Pow(S), which expresses that the intersection of an open set with a certain closed

set is inhabited. We shall call formal topologies with these extra structures open and

balanced respectively. Unary positivity predicates have been in use since [73], and

are related to the notion of an open locale, but the addition of a binary positivity

predicate in [75] is a relatively new idea that is not yet fully understood. We shall

explore some aspects of balanced formal topologies in Chapter 7.

In [7], functors between categories of constructive topological spaces and formal

topologies are defined, similar to the functors Ω and Pt between the categories Top

and Loc. The link between formal topology and locale theory is made explicit by

constructing an equivalence between the category of formal topologies and that of


set-generated Locales, that is, “locales” whose elements may form a proper class, but

which have a base that is a set. Both the adjunction and the equivalence will be

generalised in Chapter 3 to include open and balanced formal topologies.

An important class of formal topologies is that of set-presentable (or inductively

generated) formal topologies. While the category of locales has all small limits, small

colimits and function-spaces, many of these constructions do not appear to be possible

in the category of formal topologies (see [62]). However, if the covering relation / is

generated by a covering system (effectively a set of relations on the generating set

S), then constructions such as cartesian products, equalisers and function-spaces can

be carried out predicatively. A survey of some of the constructions which are made

possible by the use of set-presentable formal topologies is given in Chapter 4.

1.2.3 Point-free constructivisation of classical mathematics

One of the main uses of formal topology is to reformulate definitions and results from

classical mathematics, replacing the standard point-set notions with the correspond-

ing point-free notions. Results concerning particular sober topological spaces, for

example the real numbers, can be constructivised by substituting a suitable formal

topology for the space.

Suppose that X is a constructive topological space (see Chapter 2) and that we

have a classical theorem which says that X has some property, for example compact-

ness, that we are unable to prove constructively. If X is sober then up to homeo-

morphism X is the space of formal points of some formal topology, namely S = ΩX .

There is a definition of compactness for formal topologies, and X is compact if and

only if S is compact. However, we may be able to find another formal topology S ′

whose space of formal points is X , such that S ′ is constructively compact. Even

though this does not imply that X itself is compact, it could be possible to deduce

some other properties of X from this which would normally be proved using the com-

pactness of X . Often the formal topology S ′ can be chosen so that S = S ′ in ZFC,

so the constructive point-free theorem that has been proved is classically equivalent


to the point-sensitive result.

An interesting example of this method can be found in [24], which gives a construc-

tive proof of a point-free version of the infinite Ramsey theorem. Although the result

is described in terms of transition systems, it can easily be reinterpreted using induc-

tively generated formal topologies (see Section 4.3). The point-free infinite Ramsey

theorem is used to deduce two constructive point-sensitive results which normally

follow from the classical point-sensitive infinite Ramsey theorem: the finite Ramsey

theorem and the Paris-Harrington statement. Another example is the point-free proof

of the Open Induction Principle [22], which will be analysed in Section 4.3.

Connection with Bishop-style mathematics

Bishop avoided the use of general topology in his constructivism, preferring instead to

concentrate on more concrete structures such as the real numbers, or metric spaces.

The reason for this is that he found it necessary to use alternative definitions of

some topological concepts which, while they are classically equivalent to the standard

definitions, are not the same constructively. Two examples already mentioned above

are the functions R → R, which are defined to be continuous if and only if they are

uniformly continuous on every closed bounded interval, and metric spaces, which are

defined to be compact if and only if they are complete and totally bounded. Neither

of these definitions can be extended in a natural way to general topological spaces,

and both differ constructively from the standard topological definitions.

Formal topology can be seen as an alternative framework for constructive general

topology which is consistent with Bishop-style constructivism. For example, if we

replace the topological space of real numbers with a certain inductively generated

formal topology R whose space of formal points is homeomorphic to R, then the

formal topology mapsR→ R correspond to the functions R → R which are uniformly

continuous on every closed bounded interval (see [64]). Similarly if metric (or more

generally uniform) formal topologies are used in place of metric spaces, we have the

result that a metric (or uniform) formal topology is totally bounded if and only if its


point-free completion is compact.

1.3 Overview

As the title suggests, the aim of the thesis is to make a contribution to both point-

set and point-free topology in the constructive, predicative set theory CZF (or an

extension such as CZF+REA). When working constructively, it is often easier to

prove a point-free version of a result than the standard point-sensitive version. As

was mentioned in the previous section, the point-free results can sometimes be used to

prove facts about some topological spaces, and for this reason we shall pay particular

attention to the relationship between the point-sensitive and point-free versions of

the various topological notions considered.

One distinctive feature of this thesis is that we work in CZF, as opposed to much of

the other literature on formal topology which uses ITT. Because of this, any instances

where we can use the Strong and Subset Collection schemes as an alternative to type-

theoretic choice will be of particular interest to us.

1.3.1 Plan of the thesis

In Chapter 2 we introduce the notion of a constructive topological space (or ct-space)

from [7], and describe various constructive versions of the separation axioms Ti for

i = 0, 1, 2, 3, and of regularity. We introduce the notion of a weakly sober space,

which we first defined in [8].

Chapter 3 describes the Galois adjunction between certain categories of ct-spaces

and formal topologies, and the equivalence between categories of formal topologies

and set-generated Locales, both of which come from [7]. We generalise these results

to include open and balanced formal topologies. We also consider the new notion

of a quasi-formal topology. The chapter concludes with some definitions concerning

subspaces and separation properties of formal topologies.

In Chapter 4 we consider the class of inductively generated formal topologies.

A summary of many of the known constructions which are possible on inductively


generated formal topologies is given, and we investigate the construction of weakly

closed subspaces of inductively generated formal topologies. Class-covering systems

are introduced to allow the inductive generation of quasi-formal topologies, and we

show how the coinductive generation of a binary positivity predicate [84] can be

carried out in an extension of CZF.

Chapters 5, 6 cover the constructive theory of metric and uniform spaces in both

a point-free and point-sensitive setting. In both chapters we begin by describing the

spaces, then move on to describe the extra structure which must be added to a formal

topology to model these spaces in a point-free way, and describe the relationship

between the point-sensitive and point-free versions of the spaces. We also describe

the point-free completion of a metric or uniform formal topology, and prove a point-

free result linking total boundedness to compactness in metric and uniform formal

topology completions. As metric formal topologies are a special case of uniform formal

topologies, some of the details of the proofs in Chapter 5 are left until Chapter 6.

Chapter 7 examines some aspects of the Basic Picture in an attempt to discover

the difference between balanced formal topologies and “ordinary” formal topologies

or locales, working either in CZF, IZF or classical ZF set theory.

1.3.2 Summary of new results

The main contributions of the thesis fall into three main strands:

• Formal topology (Chapters 2, 3 and 4)

• Metric and uniform spaces and formal topologies (Chapters 5 and 6)

• Binary positivity predicates and the Basic Picture (Chapter 7 and parts of

Chapters 2, 3 and 4)

Formal topology

One new concept introduced is that of a quasi-formal topology. These are likely to

be useful for two reasons: firstly, every quasi-formal topology can be inductively


generated by a class-covering system, in contrast to ordinary formal topologies and

covering systems, and secondly quasi-formal topologies can be inductively generated

without the need for REA.

Weakly closed sublocales and the corresponding notion of weak regularity have

been known about in Locale Theory for a number of years (see, for example, [47, 48]),

but until recently there has been no attempt to formulate them in predicative formal

topology. Some of our work on this subject, particularly on weakly closed sublocales

of inductively generated formal topologies, has duplicated that of Vickers in [93],

and in both cases the initial motivation was to try to gain a better understanding of

balanced formal topologies. The approach taken here differs slightly from [93], and

we begin by discussing the more general problem of defining the weak closure of a

subspace of an arbitrary formal topology. As an application, we show how weakly

closed subspaces can be used to provide a better understanding of Coquand’s point-

free version of the Open Induction Principle [22, 25].

Having defined weak closure, we are able to formulate a predicative definition of

a weakly regular formal topology, based on [48]. In this thesis we refer to weakly

regular and regular formal topologies as “1-regular” and “3-regular” respectively,

and introduce another form of regularity which we call “2-regularity”. 2-regularity

appears to be more useful constructively than 1-regularity, but we have so far been

unable to prove whether or not 1- and 2-regularity are equivalent.

Chapters 3 and 4 also make several less significant contributions, including the

Galois adjunction and equivalence for open formal topologies, and the proof in CZF

that set-presentability is invariant under isomorphisms of formal topologies (using

Subset Collection and Strong Collection instead of type-theoretic choice, which was

used in [33]).

Metric and uniform spaces and formal topologies

We introduce the notion of a basic diameter in Chapter 5, allowing metric spaces to

be formulated predicatively in a point-free way. Prior to this, the only predicative


treatment of metric formal topologies has been in [29, 33], where a notion of an

elementary diameter was used instead. Our approach makes the connection with

Pultr’s metric locales clearer, and avoids the need to use “chains” of overlapping

basic opens to compute distances.

The correspondence between gauges and proper preuniformities in Section 6.4 is

straightforward in the impredicative intuitionistic case, as the classical version from

[67] can easily be adapted to avoid the use of classical logic. Working predicatively in

CZF, the proof is slightly harder; we have to use Strong Collection to collect enough

basic diameters to form a suitable gauge for a given preuniformity.

The constructivisation of Krız’s uniform locale completions [53] is almost certainly

new, as very little has been published on the constructive theory of uniform locales

(predicative or otherwise). The two main points worth noting are the fact that a

uniform formal topology and its completion can both have the same underlying set

S, provided the original space has an adequate base, and secondly the use of 2-

regularity in proving the universal property of the completion (regularity was used

in [53], which is not constructively true of all uniform formal topologies, and weak

regularity does not appear to be sufficient).

Some other minor contributions include a result relating compactness and total

boundedness of uniform formal topology completions, proved using the upper pow-

erlocale (an easy generalisation of a result from [92] involving point-free completions

of metric spaces); subspaces of metric formal topologies, and the observation that a

weakly closed subspace of a metric formal topology is complete.

Binary positivity predicates and the Basic Picture

The work of Sambin et al. in this field [75, 74, 76, 78, 77] has focused mainly on

basic pairs (a more general notion than that of a topological space), and continuous

relations between them. Sambin’s point-free version of a basic pair is a basic formal

topology : a formal topology to which a binary positivity predicate n has been added,

but which is only required to satisfy the first two of the three axioms A1–A3 defining


a formal topology. We shall mainly restrict our attention to topological spaces and

continuous functions, and to balanced formal topologies (basic formal topologies sat-

isfying A3). These have also been considered by Sambin et al., but in less depth than

basic pairs and basic formal topologies. Our main aim is to try to discover something

about how balanced formal topologies are related to locales and formal topologies,

not just in CZF but also in IZF and ZF.

Sambin has already explained how to obtain a balanced formal topology from a

topological space and vice-versa. In Chapter 3 we take this one step further and

obtain a Galois adjunction between categories of topological spaces and balanced

formal topologies, analogous to the one between spaces and locales. One of the most

interesting results to emerge from this is the constructive notion of weak sobriety, and

the fact that every T2 space is weakly sober. We first introduced this notion in [8],

and it is defined here in Chapter 2. Also contained in Chapter 3 is an equivalence

in IZF between the category of balanced formal topologies and that of locales with

formal closed sets (or FC-locales), which may be of interest to some locale-theorists

as it goes some way towards clarifying the difference between locales and balanced

formal topologies. In Chapter 7 we show how limits and colimits can be constructed

in the category of FC-locales, and describe some of the functors that exist between

the categories of locales and FC-locales.

Adding a binary positivity predicate to a formal topology does not merely provide

extra information about the topology. Changing the positivity predicate can have an

effect on the formal topology, even classically. In particular making n smaller gives

a balanced formal topology fewer points. Classically the “best” binary positivity

predicate is given by a n U iff ¬(a / UC), where UC is the complement of U , but

others are possible, for example ¬(a n U) for all a and U . We describe two ways in

which one might try to add new axioms to ensure that classically n is determined by

/, and explain why neither is satisfactory from a constructive point of view.

We end by giving two counterexamples to an open problem mentioned in [76],

concerning continuous relations between basic pairs.

Chapter 2

Constructive Topological Spaces

2.1 Basic pairs and ct-spaces

The standard, impredicative definition of a topological space (that is, a setX together

with a set of “open” subsets of X) cannot be used in CZF because the open sets of

an inhabited space cannot form a set. To see this, if X is an inhabited topological

space, then Up = x ∈ X | 0 ∈ p is an open set for all p ∈ Pow(0), and p 7→ Up

is a bijection between Pow(0) and the class U = Up | p ∈ Pow(0). If the open

subsets of X formed a set OX, then we could show that U = U ∈ OX | (∀x, y ∈

X)[x ∈ U ↔ y ∈ U ], so U is a set by Restricted Separation and hence Pow(0) is

a set.

Rather than simply generalise the definition to allow for a class of open sets, a

more useful definition is obtained by requiring the existence of a set-indexed family

of basic open sets (or, more generally, of basic open classes). This gives rise to the

notion of a ct-space, defined later in this section. However, we shall first consider the

more general notion of a basic pair, which was introduced by Sambin and Gebellato

in [75, 74, 72].

28

CHAPTER 2. CONSTRUCTIVE TOPOLOGICAL SPACES 29

2.1.1 Closure and interior operators arising from binary Re-

lations

Definition 2.1.1 Let X be a class, and suppose that cl(Y ) is a subclass of X for

each subclass Y of X. We say that cl is a closure operator on the subclasses of X if,

for any subclasses Y, Z ⊆ X, we can prove that

(i) cl(Y ) ⊆ cl(Z) whenever Y ⊆ Z

(ii) Y ⊆ cl(cl(Y )) ⊆ cl(Y )

Definition 2.1.2 If X is a class and int(Y ) is a subclass of X for each subclass

Y ⊆ X, we say that int is an interior operator on the subclasses of X if, for any

subclasses Y, Z ⊆ X, we can prove that

(i) int(Y ) ⊆ int(Z) whenever Y ⊆ Z

(ii) int(Y ) ⊆ int(int(Y )) ⊆ Y

Given a Relation r ⊆ A × B between two classes, we can define four operations

on the subclasses of A and B. Given U ⊆ A, define

rU = b ∈ B | (∃a ∈ A)[a ∈ U & arb]

and r−∗U = b ∈ B | (∀a ∈ A)[arb→ a ∈ U ]

In the other direction, given V ⊆ B we can define

r−V = a ∈ A | (∃b ∈ B)[b ∈ V & arb]

and r∗V = a ∈ A | (∀b ∈ B)[arb→ b ∈ V ]

The following result was proved in [74] for relations between sets, and can be gener-

alised to cover Relations between classes with no modification to the proof:

Proposition 2.1.3 Given a binary Relation r ⊆ A × B, the operations r, r−∗, r−

and r∗ are monotone w.r.t. the ordering of the subclasses of A and B by inclusion.

There are adjunctions r a r∗ and r− a r−∗, that is,

U ⊆ r∗V ⇔ rU ⊆ V and V ⊆ r−∗U ⇔ r−V ⊆ U


for any subclasses U ⊆ A and V ⊆ B. It follows that

rr∗rU = rU r−r−∗r−V = r−V

r∗rr∗V = rV r−∗r−r−∗U = r−∗U

for any U ⊆ A and V ⊆ B, and also that r∗r and r−∗r− are closure operators on the

subclasses of A and B respectively, and that r−r−∗ and rr∗ are interior operators on

the subclasses of A and B.

2.1.2 Basic pairs

Definition 2.1.4 A basic pair X consists of two classes X,S and a Relation

⊆ X × S. Informally we shall write X = 〈X, , S〉, although we have not defined

ordered triples of classes in CZF.

If X, S and are sets, we shall say that X is small1.

The intended meaning is that X is the class of points of a generalised topological

space, S is the class of “names” of certain basic open classes, and x a means that

the point x lies in the basic open class whose “name” is a.

Let ext and rest be the operators ( )− and ( )∗ respectively from subclasses of S

to subclasses of X, and let ♦ and be the operators ( ) and ( )−∗ from subclasses

of X to subclasses of S. For x ∈ X and a ∈ S, write ♦x for ♦x, and write ext a for

exta. So we have:

extU = x ∈ X | (∃a ∈ U)[x a]

♦Y = a ∈ S | ext a G Y

restU = x ∈ X |♦x ⊆ U

Y = a ∈ S | ext a ⊆ Y

where A G B means that A ∩B is inhabited. For each subclass Y ⊆ X, let

intY = ext Y = x ∈ X | (∃a ∈ S)[x ∈ ext a & ext a ⊆ Y ]

and clY = rest ♦Y = x ∈ X | (∀a ∈ S)[x ∈ ext a→ ext a G Y ]

1Sambin and Gebellato’s definition of a basic pair is the same as our definition of a small basicpair.


By Proposition 2.1.3, int is an interior operator and cl is a closure operator on the

subclasses of X. We say that a subclass Y ⊆ X is open if Y = intY , or equivalently if

Y = extU for some subclass U ⊆ S, and that Y is closed if Y = clY , or equivalently

Y = restU for some subclass U ⊆ S. This notion of closedness is one of several

possible constructive definitions (the most common alternative being the complement

of an open set), and appears to be the most useful.

Similarly we can define a closure operator and an interior operator on the sub-

classes of S. For each subclass U ⊆ S, let

AU = extU = a ∈ S | ext a ⊆ extU

and JU = ♦ restU = a ∈ S | ext a G restU

A is a closure operator and J is an interior operator; we call A and J the saturation

and reduction operators of X respectively.

We shall say that U ⊆ S is formal open if U = AU , and formal closed if U = JU .

The open and closed subclasses of X are sometimes referred to as the concrete open

and closed subclasses. It is shown in [74] that the operators ext and establish a

one-one correspondence between the concrete open subclasses of X and the formal

open subclasses of S, and that the operators rest and ♦ give a one-one correspondence

between the concrete closed and formal closed subclasses of X and S respectively.

2.1.3 Constructive topological spaces

In a basic pair X = 〈X, , S〉, the union of a collection of open classes is always

open; however, to obtain a constructive definition of a topological space, we need to

add axioms to ensure that finite intersections of open classes are open. In addition,

allowing both X and S to be proper classes is too general to be able to move easily

from topological spaces to formal topologies (covered in the next chapter), so we will

require S to be a set, and impose some restrictions on the size of the Relation .

The following definition appears in [7] and [8].

Definition 2.1.5 A constructive topological space, or ct-space, is a basic pair X =

〈X, , S〉 such that S is a set and


CS1: X = extS

CS2: ext a ∩ ext b is open for all a, b ∈ S

CS3: αx and y ∈ X |αy = αx are sets for all x ∈ X

where αx = ♦x = a ∈ S |x a.

Small and quasi-small spaces

A ct-space X = 〈X, , S〉 is small if X is a set. If this is the case, then it follows from

CS3 that is a set, and so extU , restU , Y and ♦Y are sets for all U ∈ Pow(S)

and Y ∈ Pow(X).

More generally, X is quasi-small if X has a full subset, that is, a subset Y ⊆ X

such that

(∀x ∈ X)(∀a ∈ αx)(∃y ∈ Y )[a ∈ αy & αy ⊆ αx]

Note that every full subset is dense. Moreover, a subset Y ⊆ X is full if and only if

every point x ∈ X lies in the closure of the set of points y ∈ Y which are below x in

the specialisation order.

Proposition 2.1.6 If X = 〈X, , S〉 is quasi-small, then AU and JU are sets for

all U ∈ Pow(S).

Proof. If Y is a full subset of X, then it is easy to check that for all U ∈ Pow(S),

AU = a ∈ S | (∀y ∈ Y )[a ∈ αy → αy G U ]

and JU = a ∈ S | (∃y ∈ Y )[a ∈ αy & αy ⊆ U ]

both of which are sets by Restricted Separation.

The category of ct-spaces

Given two ct-spaces X = 〈X, , S〉 and X ′ = 〈X ′, ′, S ′〉, a Function f : X → Y is a

continuous map from X to X ′ if f−1(ext′ a′) is an open subclass of X for all a′ ∈ S ′.

The ct-spaces and continuous maps form a category, Top.


2.2 Constructive separation axioms

In classical topology, there is a wide variety of separation axioms and other proper-

ties that can be satisfied by topological spaces. These are listed in [81], where the

implications among them are described in some detail, and a comprehensive list of

counterexamples to many of the other implications is given.

Constructively the situation is somewhat more complicated, as there can be sev-

eral classically equivalent ways of defining the same property which cannot be proved

to be equivalent constructively. In [8], the separation axioms T0, T1, T2 and T3 were

considered, together with the notion of a regular space. For each axiom, three clas-

sically equivalent forms Ti, T+i and T ]

i were defined. The Ti and T ]i forms were

originally defined in [40], and the T+i forms were formulated in [8] based on some of

the ideas in [17]. The main findings of [8] concerning these separation axioms are

summarised below.

2.2.1 Constructive forms of the axioms T0, T1 and T2

For i = 0, 1, 2, define a binary Relation ∼i on the points of a ct-space X as follows.

• x ∼0 y iff αx = αy

• x ∼1 y iff αx ⊆ αy

• x ∼2 y iff (∀a ∈ αx)(∀b ∈ αy)[ext a G ext b]

Definition 2.2.1 For i = 0, 1, 2, a ct-space X is Ti if, for all x, y ∈ X,

(x ∼i y) ⇒ (x = y)

To define the stronger forms T+i and T ]

i of the properties, we need some notions

of inequality between points. For all x, y ∈ X, let

• x 6=0 y iff (∃a ∈ αx)[a /∈ αy] ∨ (∃b ∈ αy)[b /∈ αx]

• x 6=1 y iff (∃a ∈ αx)[a /∈ αy]


• x 6=2 y iff (∃a ∈ αx)(∃b ∈ αy)[ext a ∩ ext b = ∅]

The inequality Relations 6=i express that x and y can be “separated” from each

other using the open sets of the ct-space. Note that (x 6=i y) ⇒ ¬(x ∼i y) for each i.

Classically the converse also holds, so that the definitions of Ti given above can be seen

to be equivalent to the standard notions, which are of the form ¬(x = y) ⇒ (x 6=i y).

Definition 2.2.2 For i = 0, 1, 2, a ct-space X is T+i if, for all x, y ∈ X,

¬(x 6=i y) ⇒ (x = y)

X is T ]i if it is T0 and, for all x, y ∈ X,

(∀a ∈ αx)[y 6=i x ∨ a ∈ αy]

2.2.2 Constructive notions of regularity and T3

As with the axioms Ti, three different constructive notions of regularity can be de-

fined; these are denoted by R, R+ and R].

Definition 2.2.3 Given a ct-space X = 〈X, , S〉,

• X is R if

(∀x ∈ X)(∀a ∈ αx)(∃b ∈ αx)[cl(ext b) ⊆ ext a]

• X is R+ if

(∀x ∈ X)(∀a ∈ αx)(∃b ∈ αx)[wcl(ext b) ⊆ ext a]

where wcl(Y ) = x ∈ X | (∀a ∈ αx)¬¬[ext a G Y ], the class of weak limit

points of Y .

• X is R] (or regular) if

(∀x ∈ X)(∀a ∈ αx)(∃b ∈ αx)[X ⊆ ext a ∪ int(X\ ext b)]

Definition 2.2.4 A ct-space X is T3 (resp. T+3 or T ]

3) if and only if it is R (resp.

R+ or R]) and T0.


2.2.3 Some implications and counterexamples

In [8] it was shown that the following implications hold between the separation ax-

ioms:

T ]3 ==⇒ T ]

2 ==⇒ T ]1 ==⇒ T ]

0

T+3

www==⇒ T+

2

www==⇒ T+

1

www==⇒ T+

0

www

T3

wwww===⇒ T2

wwww===⇒ T1

wwww===⇒ T0

wwwwIt was also shown in [8] that the standard counterexamples to the converse im-

plications Ti ⇒ Ti+1 from [81] can be constructed as (small) ct-spaces in CZF, and

that these provide constructive examples of ct-spaces which are T ]i but not Ti+1 for

i = 0, 1, 2.

Weak counterexamples to the implications Ti ⇒ T+i and T+

i ⇒ T ]i can be provided

by the discrete topology on a set.

Definition 2.2.5 Given a set X, the discrete ct-space on X is the small ct-space

X = 〈X, , X〉, where x y iff x = y.

Given Y ⊆ X, extY = restY = Y and Y = ♦Y = Y , so intY = clY = Y . For

i = 0, 1, 2, x ∼i y iff x = y and x 6=i y iff ¬(x = y). The following result and its

corollary were observed in [8].

Proposition 2.2.6 Let X be the discrete ct-space on a set X. For i = 0, 1, 2, 3:

(i) X is Ti

(ii) (X is T+i ) ⇔ (X is R+) ⇔ (∀x, y ∈ X)[¬¬(x = y) → (x = y)]

(iii) (X is T ]i ) ⇔ (X is R]) ⇔ (∀x, y ∈ X)[(x = y) ∨ ¬(x = y)]

Corollary 2.2.7

(i) (Every Ti space is T+i ) ⇒ REM for i = 0, 1, 2, 3


(ii) (Every R space is R+) ⇒ REM

(iii) (Every T+i space is T ]

i ) ⇒ wREM for i = 0, 1, 2, 3

(iv) (Every R+ space is R]) ⇒ wREM

2.3 Sobriety and weak sobriety

Sobriety can be thought of as another constructive separation property. The notion

of a sober space originally arose from the adjoint pair of functors Ω and Pt between

the categories of topological spaces and locales (see, for example, [45]). A topological

space is sober if and only if, up to isomorphism, it can be obtained as the space of

points of a locale.

We will begin by giving a predicative definition of sobriety for ct-spaces, taken

from [7], where it was shown to be related to the functors Ω and Pt between certain

categories of ct-spaces and formal topologies. Classically every Hausdorff space is

sober, but this is not true constructively (see [37], [8]).

A recent development in formal topology involves the addition of a binary posi-

tivity predicate (see [75, 72, 78]). In [8] we observed that the definition of a formal

point of a balanced formal topology from [72, 78] leads to a new, constructively

weaker notion of sobriety. Classically the two notions of sobriety are equivalent, but

constructively we are able to show that every T2 space is weakly sober.

For the details of how weak sobriety is related to balanced formal topologies,

see Proposition 3.2.20. The difference (constructively) between the two notions of

sobriety plays an important part in the discussion of several issues concerning the

Basic Picture in Chapter 7.

2.3.1 Ideal points and sober spaces

Definition 2.3.1 An ideal point of a ct-space X = 〈X, , S〉 is a subset α ⊆ S such

that

1. ∃a ∈ α,


2. (∀a, b ∈ α)(∃c ∈ α)[ext c ⊆ ext a ∩ ext b],

3. (∀a ∈ α)(∀U ∈ Pow(S))[ext a ⊆ extU → α G U ].

It is easy to check that αx = a ∈ S |x a is an ideal point for each point

x ∈ X.

Definition 2.3.2 A ct-space X is sober if for each ideal point α there is a unique

point x ∈ X such that α = αx.

For each ct-space X , we can define a new ct-space Sob(X ) = 〈X ′, ′, S〉, where X ′

is the class of ideal points of X and α ′ a iff a ∈ α. Part (1) of the above definition

ensures that Sob(X ) satisfies the axiom CS1; CS2 can be proved from (2) and (3),

and CS3 follows from the definition of ′.

Sob(X ) is known as the soberification of X . It can be shown that the ideal points

of X are precisely the ideal points of Sob(X ), so the soberification of a ct-space is

always sober.

Proposition 2.3.3 For any ct-space X , AU = A′U for all U ∈ Pow(S), where A

and A′ are the saturation operators of X and Sob(X ) respectively.

Proof. A′U ⊆ AU follows from the fact that αx is an ideal point for all x ∈ X. The

converse, AU ⊆ A′U , is a consequence of part (3) of Definition 2.3.1.

Proposition 2.3.4 Given a ct-space X , the mapping x 7→ αx defines a continuous

function ηX : X → Sob(X ).

ηX is a homeomorphism (that is, an isomorphism in Top) if and only if X is

sober.

Proof. Given a ∈ S, the preimage of the basic open ext′ a is η−1X (ext′ a) = x ∈

X | a ∈ αx = ext a, which is open in X . So ηX is continuous.

The second part amounts to showing that ηX is a homeomorphism if and only if

it is a bijection. Suppose that ηX is a bijection. We have to show that its inverse


is continuous. But if a ∈ S then ηX [ext a] = αx |x ∈ ext a = α ∈ X ′ | a ∈ α =

ext′ a.

It follows from the “uniqueness” part of the definition of sobriety that every sober

space must be T0. Classically it can be shown that every T2 space is sober (see, for

example, [45]. However the proof is non-constructive, and it was shown in [37] using

sheaf models that it cannot be shown intuitionistically that the space of rational

numbers is sober. Further to this, in [8] we proved the following:

Theorem 2.3.5 In CZF, the rational numbers with the Euclidean topology form a

T2 ct-space. If this space is sober then the non-constructive principle LPO holds.

2.3.2 Strong ideal points and weak sobriety

Definition 2.3.6 A strong ideal point of a ct-space X = 〈X, , S〉 is a subset α ⊆ S

such that

1. ∃a ∈ α

2. (∀a, b ∈ α)(∃c ∈ α)[ext c ⊆ ext a ∩ ext b]

3. (∀a ∈ α)(∃x ∈ ext a)[αx ⊆ α].

Conditions (1) and (2) are identical to those from Definition 2.3.1. (3) in this

definition implies (3) from Definition 2.3.1 constructively, and classically the two are

equivalent. Again it can be shown that αx is a strong ideal point for each point

x ∈ X.

Definition 2.3.7 A ct-space X is weakly sober if for each strong ideal point α there

is a unique point x ∈ X such that α = αx.

For each ct-space X , let WSob(X ) = 〈X ′, ′, S〉, where X ′ is the class of strong

ideal points of X and α ′ a if and only if a ∈ α. WSob(X ) is the weak soberification

of X . The strong ideal points of WSob(X ) are precisely the strong ideal points of X ,

so the weak soberification of a ct-space is always weakly sober.


Proposition 2.3.8 Given a ct-space X , AU = A′U and JU = J ′U for all U ∈

Pow(S), where A and J are the saturation and reduction operators of X , and A′

and J ′ are the saturation and reduction operators of WSob(X ).

Proof. The fact that AU = A′U follows from the proof of Proposition 2.3.3, since

every strong ideal point is an ideal point.

Suppose that a ∈ JU . There is an x ∈ ext a such that αx ⊆ U . αx ∈ ext′ a∩rest′ U

so a ∈ J ′U .

Now suppose that a ∈ J ′U . There is a strong ideal point α such that a ∈ α and

α ⊆ U . By part (3) of Definition 2.3.6 there is an x ∈ ext a such that αx ⊆ α, so

αx ⊆ U and hence a ∈ JU .

Proposition 2.3.9 Given a ct-space X , the mapping x 7→ αx defines a continuous

function ηX : X → WSob(X ).

ηX is a homeomorphism if and only if X is weakly sober.

Proof. The proof is identical to that of Proposition 2.3.4.

As with sobriety, it follows from the “uniqueness” part of the definition of weak

sobriety that every weakly sober space is T0. Furthermore we have the following

result, which we proved in [8].

Proposition 2.3.10 Every T2 ct-space is weakly sober.

Remark. Part (3) of the definition of a weak ideal point can equivalently be written

as α = Jα, that is, α is a formal closed set in X . If X is a small ct-space, then

under the correspondence between concrete closed and formal closed sets, the weak

ideal points correspond to inhabited, closed subsets F of X such that

(∀a, b ∈ S)[ext a G F & ext b G F → (ext a ∩ ext b) G F ]

In the presence of classical logic, this is the same as the notion of an irreducible closed

subset used in the characterisation of sobriety in [45].

Chapter 3

Formal Topology

3.1 Formal topology

We begin the chapter by defining the notion of a formal topology. The particular

definition used here is taken from [7], but does not differ significantly from the other

variants. As in [7], we have included a preorder ≤ in the definition; this is not present

in some other definitions, and makes little difference to the theory, but it does make

some subjects such as subspaces and the inductive generation of formal topologies

slightly simpler.

Two other categories of structures, known as open and balanced formal topologies,

will be defined in Section 3.2, where we also show how to define a Galois adjunction

between each kind of formal topology and the category of ct-spaces, based on the

method in [7]. In Section 3.3 we construct an equivalence between each category of

formal topologies and a certain category of Frames, with some added structure in

the case of balanced formal topologies. Section 3.4 describes a generalisation of the

notion of a formal topology, in which a certain smallness condition has been omitted.

In Sections 3.5 and 3.6 we build on the work of Curi in [33], considering the notion

of a subspace of a formal topology, and various separation properties. In particular

we examine the problem of defining weakly closed and strongly dense subspaces, and

consider the related notion of weak regularity.

40

CHAPTER 3. FORMAL TOPOLOGY 41

Definition 3.1.1 A formal topology is a set S equipped with a preorder ≤ and a

Relation / ⊆ S×Pow(S) such that AU = a ∈ S | a/U is a set for each U ∈ Pow(S),

and

A1: ↓U ⊆ AU

A2: U ⊆ AV ⇒ AU ⊆ AV

A3: AU ∩ AV ⊆ A(U ↓V )

where ↓U = a ∈ S | (∃b ∈ U)[a ≤ b] and U ↓V = (↓U) ∩ (↓V ).

The Relation / and the operator A on Pow(S) are known as the covering relation

and the saturation operator of the formal topology.

We shall use calligraphic letters S, T , . . . to denote formal topologies (including

the set S together with the structure given by ≤ and /). Unless otherwise stated it

should be assumed that any subscripts, superscripts, primes etc. added to the name

of a formal topology match those of its underlying set, preorder, covering relation

and saturation operator. For example, if we say that S1 is a formal topology, then

its underlying set will be S1, its preorder and covering relation will be ≤1 and /1

respectively and its saturation operator will be A1.

If U, V ∈ Pow(S), let U/V abbreviate the formula U ⊆ AV . We say that a subset

U ∈ Pow(S) is saturated if AU = U . Let Sat(S) be the class of saturated subsets

of S. It can be shown that Sat(S) is a Frame for every formal topology S (that is,

it is a partially ordered class with finite meets and joins of all subsets, satisfying the

infinite distributive law).

Definition 3.1.2 If S and S ′ are formal topologies, a formal topology map from S

to S ′ is a relation r ⊆ S × S ′ such that

FTM1: Ar−S ′ = S

FTM2: r−a′ ∩ r−b′ ⊆ Ar−(a′ ↓b′) for all a′, b′ ∈ S ′

FTM3: r−A′U ′ ⊆ Ar−U ′ for all U ′ ∈ Pow(S ′)


FTM4: r−a′ = Ar−a′ for all a′ ∈ S ′

To see the connection with locale theory, note that r determines a Frame mor-

phism from Sat(S ′) to Sat(S) given by U ′ 7→ Ar−U . FTM1, FTM2 and FTM3

ensure that r preserves >, ∧ and∨

respectively. FTM4 is included so that any two

formal topology maps which give rise to the same Frame homomorphism must be

extensionally equal. In some of the literature (e.g. [33]), axioms similar to FTM1-3

are used, and two maps r1, r2 are defined to be equal when r−1 a′ =S r−2 a′ for all

a′ ∈ S. The approach taken here of using the equality axiom FTM4 is the one taken

in [7].

Proposition 3.1.3 The formal topologies and formal topology maps form a category,

which we shall call FTop.

Proof. The identity map on a formal topology S is given by

idS = 〈a, b〉 ∈ S × S | a ∈ Ab

and the composite of r : S → S ′ and r′ : S ′ → S ′′ is defined by

r′ r = 〈a, a′′〉 ∈ S × S ′′ | a ∈ Ar−r′−a′′

It is easy to check that these are formal topology maps and that they satisfy the

axioms for a category.

The following proposition gives some useful axioms which are equivalent to FTM3.

A more extensive list is given by Sambin and Gebellato in [77].

Proposition 3.1.4 If S and S ′ are formal topologies and r ⊆ S × S ′, the following

are equivalent:

1. r−A′U ′ ⊆ Ar−U ′ for all U ′ ∈ Pow(S ′) (FTM3)

2. ara′ & a′ /′ U ′ ⇒ a / r−U ′ for all a ∈ S, a′ ∈ S ′ and U ′ ∈ Pow(S ′)

3. U = AU ⇒ r−∗U = A′r−∗U for all U ∈ Pow(S)


Proof. See [77].

It is well-known that two frames are isomorphic if and only if there is an order-

preserving bijection from one to the other whose inverse is also order-preserving; such

a bijection will always preserve 1, ∧ and∨

. Similarly for formal topologies we have

the following result.

Proposition 3.1.5 Let S and S ′ be formal topologies, and suppose that r ⊆ S × S ′

and s ⊆ S ′ × S are two relations, each satisfying FTM3 and FTM4, such that

• r−s−a =S a for all a ∈ S

• s−r−a′ =S′ a′ for all a′ ∈ S ′

Then r : S → S ′ and s : S ′ → S are formal topology maps, and they form an

isomorphism between S and S ′.

Proof. We will show that r satisfies FTM1 and FTM2. By symmetry the same is

true of s, and it is clear from the two conditions above that s r and r s are the

identity maps on S and S ′ respectively.

FTM1: If a ∈ S then a / r−s−a ⊆ r−S.

FTM2: Let a′, b′ ∈ S ′ and suppose that c ∈ r−a′ ∩ r−b′. Then s−c ⊆

s−r−a′ /′ a′ and s−c ⊆ s−r−b′ /′ b′, so s−c /′ a′ ↓ b′. Hence

c / r−s−c / r−(a′ ↓b′).

3.2 Three Galois adjunctions in CZF

The aim of this section is to present three Galois adjunctions in CZF, similar to the

adjunction in IZF between the category of topological spaces and the category of

locales.

The first of these adjunctions was presented in [7]. In order to obtain a formal

topology from a ct-space X , the space must satisfy certain smallness conditions to


ensure that AU is a set for each U ⊆ S. Such a space will be called a standard

ct-space. Although every formal topology S gives rise to a ct-space, this space is

not necessarily standard. In [7], this problem is solved by defining a standard formal

topology to be one whose space of formal points is standard. Every standard ct-space

can be shown to give rise to a standard formal topology; an adjunction between the

categories of standard ct-spaces and standard formal topologies can then be obtained.

We shall begin by giving a sketch of the proof of this adjunction from [7]. We

will also consider two other categories: those of open formal topologies and balanced

formal topologies. The objects of these categories are formal topologies with some

extra structure. In each case if the formal topology arises from a ct-space then the

extra structure can also be obtained from that space, provided it satisfies some extra

smallness conditions. We will show that the technique from [7] can be applied in

these cases to obtain Galois adjunctions between categories of “open-standard” ct-

spaces and “standard” open formal topologies, and between categories of “balanced-

standard” ct-spaces and “standard” balanced formal topologies.

Finally we should remark that, as in [7], in defining the various notions of stan-

dard spaces and standard formal topologies, we have attempted to find adjunctions

between the largest possible subcategories. In the case of ordinary and open formal

topologies, assuming REA, the adjoint pairs of functors can be restricted to the sub-

categories of quasi-small ct-spaces and set-presentable formal topologies (see [7]), and

in practice this is usually all that is needed.

Galois adjunctions

Given a functor G : D → C, to construct an left adjoint to G we must define two

operations F and η• on Ob(C) such that

• if X is an object of C then FX is an object of D and ηX is a morphism

X → GFX in C

• if f : X → GY in C then there is a unique morphism f : FX → Y in D such

that f = Gf ηX .


If this is the case then F extends to a functor C → D by setting Ff = ηX′ f for each

morphism f : X → X ′ in C, and η is a natural transformation idC → GF , known as

the unit of the adjunction. The counit is the natural transformation ε : FG → idD

given by εY = idGY for each Y ∈ Ob(D). The unit and counit satisfy the triangle

identities εFX FηX = idFX and GεY ηGY = idGY for all X ∈ Ob(C) and Y ∈ Ob(D).

The notion of a Galois adjunction is described in [10]. The adjunction described

above is a Galois adjunction if ηG is a natural isomorphism, that is, ηGY : GY →

GFGY is an isomorphism for all Y ∈ Ob(D). By the second triangle identity the

inverse of ηGY must be GεY : GFGY → GY . It can be shown that ηG is a natural

isomorphism if and only if Fη is a natural isomorphism; if this is the case then the

inverse of FηX is εFX : FGFX → FX.

3.2.1 The adjunction for formal topologies

From ct-spaces to formal topologies

If X = 〈X, , S〉 is a ct-space, we can preorder S by a ≤ b⇔ ext a ⊆ ext b. For each

a ∈ S and U ⊆ S, let

a / U ⇔ ext a ⊆ extU

The saturation operator A on Pow(S) obtained from / agrees with the saturation

operator of X . We shall call X a standard space if AU is a set for each subset U of

S. If this is the case, then S, ≤ and / form a formal topology, which we shall call

ΩX .

If f : X → X ′ is a continuous map between standard spaces, we say that f is a

standard continuous map if f−1(ext a′) is a set for all a′ ∈ S.

It is easy to verify that the identity map on a standard space is standard, and that

composing two standard maps gives another standard map, so the standard spaces

and standard maps form a subcategory TopS of Top.


From formal topologies to ct-spaces

Conversely, if we start with a formal topology we can obtain the ct-space of its formal

points. FTop has a terminal object 1, the formal topology on the set 1 = 0 with

AU = U for all U ⊆ 1. A formal topology map r : 1 → S is determined by the set

r0 = a ∈ S | 0 ∈ r−a. This leads to the following definition:

Definition 3.2.1 If S is a formal topology, a subset α ⊆ S is a formal point of S iff

FP1: ∃a ∈ α

FP2: (∀a, b ∈ α)(∃c ∈ α)[c ≤ a, b]

FP3: (∀a ∈ α)(∀U ∈ Pow(S))[a / U → α G U ]

Note that α is a formal point of S if and only if 〈0, a〉 | a ∈ α is a formal topology

map 1 → S. The formal points of S form a ct-space Pt(S) = 〈Pt(S), , S〉 where

Pt(S) is the class of formal points of S and α a iff a ∈ α.

Pt can be extended to a functor Pt : FTop → Top. If r : S → S ′, define

Pt(r) : Pt(S) → Pt(S ′) by

Pt(r)(α) = rα = a′ ∈ S ′ | (∃a ∈ α)[ara′]

for all α ∈ Pt(S).

As noted in [7], it does not appear to be possible to show that Pt(S) is a standard

space. The solution in the paper is to say that a formal topology S is a standard

formal topology if Pt(S) is a standard space.

Likewise we are unable to prove that Pt(r) is a standard continuous map for all

r : S → S ′. We shall say that r is a standard formal topology map if Pt(r) is a

standard continuous map.

The standard formal topologies and standard formal topology maps form a sub-

category FTopS of FTop.


Spatial formal topologies

It follows from FP3 that if a / U then a ∈ α → α G U for all α ∈ Pt(S), and so

ext a ⊆ extU in Pt(S).

Definition 3.2.2 S is spatial if (∀α ∈ Pt(S))[a ∈ α→ α G U ] implies a / U .

Proposition 3.2.3 If S is a spatial formal topology then Pt(S) is a standard ct-

space.

Proof. If A′ is the closure operator arising from the space Pt(S), then A′U = AU

and hence A′U is a set for all U ∈ Pow(S).

The adjunction

We are now ready to construct the Galois adjunction in CZF. Given a ct-space X ,

we showed in Section 2.3 that we can define a continuous map ηX : X → Sob(X ) by

ηX (x) = αx = a ∈ S |x a

for all x ∈ X.

Theorem 3.2.4 Ω and Pt can be restricted to functors ΩS : TopS → FTopS and

PtS : FTopS → TopS, and η is the unit of a Galois adjunction ΩS a PtS.

The following proposition contains everything that is required to prove Theo-

rem 3.2.4.

Proposition 3.2.5 Let X = 〈X, , S〉 be a standard ct-space.

(i) ΩX is spatial, so it is a standard formal topology.

(ii) Pt(ΩX ) = Sob(X ) and ηX : X → Sob(X ) is standard continuous.

(iii) If S ′ is a standard formal topology and g : X → Pt(S ′) is standard continuous,

then

rg = 〈a, a′〉 ∈ S × S ′ | ext a ⊆ g−1(ext′ a′)


is the unique formal topology map r : ΩX → S ′ such that g = Pt(r). Further-

more rg is a standard formal topology map.

(iv) If X = Pt(S) for some formal topology S, then X is sober.

Proof. See [7].

3.2.2 The adjunction for open formal topologies

Open formal topologies

In much of the literature on Formal Topology (see, for example, [73, 72, 27]), in

addition to the covering relation /, a formal topology also comes equipped with a

(unary) positivity predicate, that is, a subset Pos ⊆ S satisfying

Monotonicity: Pos a & a / U → PosU

Positivity: a / a | Pos a

for all a ∈ S and U ∈ Pow(S), where Pos a means a ∈ Pos and PosU means

U G Pos. The following useful facts may be easily verified (see [32]).

Proposition 3.2.6 Positivity is equivalent to the axiom

(∀a ∈ S)(∀U ∈ Pow(S))[(Pos a→ a / U) → a / U ]

In the presence of Positivity, the Monotonicity axiom is equivalent to

(∀a ∈ S)(∀U ∈ Pow(S))[Pos a & a / U → ∃b ∈ U ]

A formal topology equipped with a positivity predicate is known as an open formal

topology. This is the predicative counterpart of the notion of an open locale introduced

by Joyal and Tierney in [51] and discussed by Johnstone in [46].

The connection with open locales is covered in Section 3.3.2. We also have the

following result, which implies that the positivity predicate must be unique, if it

exists.


Proposition 3.2.7 If S is a formal topology with a positivity predicate Pos, then

PosU iff (∀V ∈ Pow(S))[U / V → ∃b ∈ V ].

Proof. See [26, Proposition 5].

The category OFTop has open formal topologies as objects, and formal topology

maps (defined in Section 3.2.1) as morphisms. By Proposition 3.2.7 we may regard

OFTop as a full subcategory of FTop.

Morphisms and points of open formal topologies

By definition, a morphism between two open formal topologies is just a formal topol-

ogy map, as defined in Definition 3.1.2, without any extra axioms concerning the

positivity predicates. However, any such morphism can be proved to have the follow-

ing property.

Proposition 3.2.8 If r : S → S ′ is a formal topology map between open formal

topologies, then Pos r−a′ ⇒ Pos′ a′ for all a′ ∈ S ′.

Proof. Suppose that Pos r−a′, so by definition there is an a ∈ r−a′ such

that Pos a. By Positivity a′ /′ a′ | Pos a′, so a / r−a′ | Pos a′. By Monotonicity

r−a′ | Pos a′ is inhabited, so Pos a′.

We have a similar result for formal points (see [60]):

Proposition 3.2.9 If α is a formal point of an open formal topology S, then α ⊆ Pos.

Proof. Let a ∈ α. Then a / a | Pos a, so by FP3 a | Pos a is inhabited and

hence Pos a.

Proposition 3.2.10 If S is a spatial formal topology with a unary positivity predicate

Pos and⋃Pt(S) is a set, then Pos a⇔ (∃α ∈ Pt(S))[a ∈ α].

Proof. The implication from right to left is just Proposition 3.2.9.

Suppose that Pos a. Let U = a ∩⋃Pt(S). Then ext a ⊆ extU in Pt(S), so

a / U . By Monotonicity U is inhabited, so there is an α ∈ Pt(S) such that a ∈ α.


The adjunction

Given a standard ct-space X = 〈X, , S〉, a natural way to obtain a positivity pred-

icate on ΩX is to take

Pos = a ∈ S | ∃x ∈ ext a

Unfortunately we are unable to show that Pos is always a set. To solve this problem,

we shall say that a ct-space is open-standard if it is standard and the class Pos defined

above is a set. The following result gives an alternative characterisation of the open-

standard ct-spaces.

Proposition 3.2.11 A standard ct-space X is open-standard if and only if it has a

dense subset, that is, a subset X0 ⊆ X such that (∀a ∈ S)[ext a G X → ext a G X0].

Proof. If X0 ⊆ X is a dense subset, then Pos a iff (∃x ∈ X0)[a ∈ αx], so Pos is a set

by restricted separation.

Conversely, suppose that Pos is a set. Then (∀a ∈ Pos)(∃x)[x ∈ X & x a], so

by Strong Collection there is a set X0 such that

(∀a ∈ Pos)(∃x ∈ X0)[x ∈ X & x a]

and (∀x ∈ X0)(∃a ∈ Pos)[x ∈ X & x a]

from which it follows that X0 is a dense subset of X.

Let ΩOX be the open formal topology arising from the open-standard ct-space X ,

with the positivity predicate Pos defined above. Let TopOS be the full subcategory

of TopS whose objects are open-standard ct-spaces.

Let PtO : OFTop → Top be the restriction of Pt to OFTop, regarded as a full

subcategory of FTop. As in the previous section, we are unable to prove that PtO(S)

is an open-standard space for any open formal topology S. We shall say that an open

formal topology is standard if PtO(S) is an open-standard space. Let OFTopS be

the category of standard open formal topologies and standard formal topology maps.

The adjunction involving open formal topologies will have the same unit as the

one for open formal topologies, that is, ηX : x 7→ αx.


Theorem 3.2.12 ΩO and PtO restrict to a pair of functors ΩOS : TopOS → OFTopS

and PtOS : OFTopS → TopOS, and η is the unit of a Galois adjunction ΩOS a PtOS.

Proof. To prove this, we must prove the analogue of Proposition 3.2.5 for open-

standard ct-spaces and standard open formal topologies. The only part which requires

any extra work is (i).

As before, if X is an open-standard ct-space then ΩOX is spatial, so X ′ =

Pt(ΩOX ) is standard. To show that X ′ is open-standard we will show that (∃α ∈

Pt(ΩOX ))[a ∈ α] iff Pos a (that is, X is dense in its soberification X ′).

If α ∈ Pt(ΩOX ) and a ∈ α, then Pos a by Proposition 3.2.9. Conversely if Pos a,

then there is an x ∈ ext a and so αx is a formal point containing a.

Thus a ∈ S | ∃α ∈ ext′ a is a set, so X ′ is open-standard.

3.2.3 The adjunction for balanced formal topologies

Balanced formal topologies

A relatively new idea in formal topology is the addition of a binary positivity predicate.

This was first introduced by Sambin and Gebellato in [75], and discussed further by

Sambin in [72] and by Sambin et al. in a series of papers on the “Basic Picture”,

including [74, 76, 78, 77]. We shall adopt the terminology used in [72], and call a

formal topology with a binary positivity predicate a balanced formal topology.

Definition 3.2.13 A balanced formal topology consists of a set S equipped with a

preorder ≤ and two Relations /,n ⊆ S × Pow(S) such that AU and JU = a ∈

S | a n U are sets for all U ∈ Pow(S), satisfying the rules A1-A3 from Definition

3.1.1, together with the following new rules:

J1: JU ⊆ U

J2: JU ⊆ V → JU ⊆ J V

C: AU G J V → U G J V


The operator J is called the reduction operator of the balanced formal topology,

and the subsets U of S such that U = JU are known as the formal closed sets of

S. Intuitively a n U means that the basic open a “touches” a certain closed set

determined by U (ext a G restU if S arises from ct-space).

Classically closed subsets can be defined as the complements of open sets, and so

n can be defined by JU = (A(UC))C . The axioms J1 and J2 are dual to A1 and A2

(it follows from J1 and C that ↑JU ⊆ U , where ↑V = a ∈ S | (∃b ∈ V )[b ≤ a]). C

is known as the Compatibility rule, and relates the closure operator A to the interior

operator J . There seems to be no obvious dual to A3 that can be added.

It is not yet clear whether any more axioms can sensibly be added to the defini-

tion of a balanced formal topology. We shall discuss this problem in some detail in

Section 7.2.

Definition 3.2.14 If S and S ′ are balanced formal topologies, a balanced formal

topology map r : S → S ′ is a relation r ⊆ S × S ′ satisfying FTM1-FTM4 from

Definition 3.1.2, together with the following extra axiom:

FTM3′: J (r∗U ′) ⊆ r∗(J ′U ′)

FTM3′ has been added based on Sambin’s definition of a “continuous relation”,

and is dual to FTM3. It follows from FTM1 and Compatibility (C) that J (r∗∅) ⊆ ∅,

which can be thought of as a dual to FTM1. There seems to be no obvious dual to

FTM2 that can be added.

In Sambin’s papers, the axiom FTM4 was omitted and two relations r1 and r2 were

defined to be “equal” if Ar−1 U ′ = Ar−2 U ′ and J r∗1U ′ = J r∗2U ′ for all U ′ ∈ Pow(S ′).

It was shown in [77] that r1 and r2 are equal if and only if Ar−1 a′ = Ar−2 a′ for

all a′ ∈ S ′. Any relation satisfying FTM1–3 and FTM3′ is “equal” to one satisfying

FTM4, so as with ordinary formal topologies we are justified in using FTM4 to ensure

that if two morphisms are “equal” then they are extensionally equal.

As with FTM3, there are a number of axioms which are equivalent to FTM3′.

Many of these are listed by Sambin and Gebellato in [77]; we will state the most

useful of them here.


Proposition 3.2.15 If S and S ′ are balanced formal topologies and r ⊆ S×S ′ then

the following are equivalent.

1. J r∗U ′ ⊆ r∗J ′U ′ for all U ′ ∈ Pow(S ′) (FTM3′)

2. an r∗U ′ & ara′ ⇒ a′ n′ U ′ for all a ∈ S and U ′ ∈ Pow(S ′)

3. U = JU ⇒ rU = J ′rU for all U ∈ Pow(S)

Proof. It is easy to see from the definitions that (1) and (2) are equivalent.

Suppose that r satisfies (1), and let U = JU . Using U ⊆ r∗rU and J2 we have

U ⊆ J r∗rU ⊆ r∗J ′rU and hence rU ⊆ J ′rU . So (3) holds.

Conversely suppose that r satisfies (3), and let U ′ ∈ Pow(S ′). Then

rJ r∗U ′ = J ′rJ r∗U ′ ⊆ J ′rr∗U ′ ⊆ J ′U ′

and hence J r∗U ′ ⊆ r∗J ′U ′.

Proposition 3.2.16 The balanced formal topologies and balanced formal topology

maps form a category BFTop. The identity map for S is

idS = 〈a, b〉 ∈ S × S | a / b

and the composite of r : S → S ′ with r′ : S ′ → S ′′ is

r′ r = 〈a, a′′〉 ∈ S × S ′′ | a / r−(r′−a′′)

Proof. It remains to prove that idS satisfies FTM3′ and that FTM3′ is preserved by

composition. We shall use (2) of Proposition 3.2.15 to prove this.

Let r = idS . If a n r∗U ′ and ara′, then J r∗U ′ G Aa′ and so a′ ∈ J r∗U ′ by

Compatibility. So idS satisfies FTM3′.

Now let r : S → S ′ and r′ : S ′ → S ′′ be balanced formal topology maps. Suppose

that an (r′ r)∗U ′′ and a (r′ r) a′′. We have to show that a′′ n′′ U ′′.

By Compatibility, there is a b ∈ S such that bn (r′ r)∗U ′′ and b ∈ r−(r′−a′′),

i.e. brb′r′a′′ for some b′ ∈ S. It is easily shown that (r′ r)∗U ′′ ⊆ r∗(r′∗U ′′), so

a n r∗(r′∗U ′′). Using FTM3′ for r gives b′ n′ r′∗U ′′, and by FTM3′ for r′ we have

a′′ n′′ U ′′.


From ct-spaces to balanced formal topologies

Given a ct-space X = 〈X, , S〉, for each U ⊆ S define / and n by

AU = extU = a ∈ S | ext a ⊆ extU

JU = ♦ restU = a ∈ S | ext a G restU

If AU and J V are sets whenever U is a set, we shall say that X is a balanced-standard

ct-space. Let TopBS be the category of balanced-standard ct-spaces and standard

continuous maps.

If X is balanced-standard, let ΩBX be the balanced formal topology on S with the

preorder a ≤ b⇔ ext a ⊆ ext b, and the Relations /,n determined by the definitions

of AU and J V above.

From balanced formal topologies to ct-spaces

The addition of the axiom FTM3′ leads to the following, stronger definition of a point

of a balanced formal topology.

Definition 3.2.17 Given a balanced formal topology S, a subset α ⊆ S is a formal

point of S if:

FP1: ∃a ∈ α

FP2: (∀a, b ∈ α)(∃c ∈ α)[c ≤ a, b]

FP3′: α = Jα

Note that by Compatibility, FP3′ implies FP3, so every formal point of a balanced

formal topology is also a formal point of the underlying “ordinary” formal topology.

Let PtB(S) be the ct-space 〈PtB(S),3, S〉, where PtB(S) is the set of formal

points of the balanced formal topology S.

PtB can also be defined on morphisms, to give a functor PtB : BFTop → Top.

If r : S → S ′ is a balanced formal topology map, for each α ∈ PtB(S) let

PtB(r)(α) = rα = a′ ∈ S ′ | (∃a ∈ α)[ara′]


The proof that PtB(r) is a continuous map PtB(S) → PtB(S ′) is identical to

the proof for Pt and ordinary formal topologies, except we must also check that

PtB(r)(α) satisfies FP3′. This follows from (3) of Proposition 3.2.15.

We say that a balanced formal topology S is standard if the ct-space PtB(S)

is balanced-standard. A balanced formal topology map r : S1 → S2 is standard if

PtB(r) is a standard continuous map. Let BFTopS be the category of standard

balanced formal topologies and standard balanced formal topology maps.

Spatial balanced formal topologies

Definition 3.2.18 A balanced formal topology S is spatial iff

• (∀α ∈ Pt(S))[a ∈ α→ α G U ] → a / U and

• an U → (∃α ∈ Pt(S))[a ∈ α & α ⊆ U ].

Note that the converses of these two implications hold for all balanced formal

topologies. In particular if α is a formal point and a ∈ α ⊆ U , then a ∈ Jα ⊆ JU

by FP3′.

If X ′ = PtB(S) and A′ and J ′ are the saturation and reduction operators of the

ct-space X ′, defined in Section 2.1, then we always have AU ⊆ A′U and J ′U ⊆ JU

for all U ∈ Pow(S). S is spatial iff AU = A′U and JU = J ′U .

Proposition 3.2.19 If S is a spatial balanced formal topology then PtB(S) is bal-

anced-standard.

Proof. Let X ′ = Pt(S) and let A′ and J ′ be the operators arising from X ′. As

noted above, if U is a set then A′U = AU and J ′U = JU , so A′U and J ′U are both

sets.

The adjunction

The unit of the adjunction will be defined by the maps ηX : X → WSob(X ), where

ηX (x) = αx.


Proposition 3.2.20 Let X = 〈X, , S〉 be a balanced-standard ct-space.

(i) ΩBX is spatial, so it is a standard balanced formal topology.

(ii) PtB(ΩBX ) = WSob(X ) and ηX : X → WSob(X ) is standard continuous.

(iii) If S ′ is a balanced formal topology and g : X → PtB(S ′) is standard continuous,

then


is the unique balanced formal topology map r : ΩBX → S ′ such that g = PtB(r).

Furthermore rg is a standard balanced formal topology map.

(iv) If X = PtB(S) for some balanced formal topology S, then X is weakly sober.

Proof. (i) We already know from Proposition 3.2.5 that (∀α ∈ Pt(S))[a ∈ α→ α G

U ] implies a / U .

Suppose that anU . Then there is a point x ∈ X such that x a and αx ⊆ U ,

and αx is a formal point of ΩBS. Thus ΩBS is spatial.

(ii) It is easy to see from the definitions that the weak ideal points of X are precisely

the formal points of ΩBX. The proof that ηX is standard continuous is identical

to the proof for Proposition 3.2.5.

(iii) First we need to check that rg is a balanced formal topology map; from Propo-

sition 3.2.5 it remains to prove that rg satisfies FTM3′.

Suppose that a n r∗gU′ and arga

′, so there is an x ∈ ext a ∩ rest(r∗gU′) and

ext a ⊆ g−1(ext′ a′). We have to show that a′ n′ U ′. Let α′ = g(x). By

continuity, for each b′ ∈ α′ there is a b ∈ αx ⊆ r∗gU′ such that ext b ⊆ g(ext b′)

and hence b rgb′, so b′ ∈ U ′. Thus α′ ⊆ U ′ and hence a′ ∈ α′ ⊆ Jα′ ⊆ JU ′.

The proof that this is the unique map r : ΩBX → S ′ such that g = PtB(r)

is similar to the proof for ordinary formal topologies. rg is standard because

PtB(rg) = g is a standard continuous map.


(iv) Let X = PtB(S). Since αα = α for every α ∈ X, to show that X is weakly sober

it suffices to show that if α is a strong ideal point of X then it is a formal point

of S.

The proofs of FP1 and FP2 are the same as in [7] and [8]. For FP3′, suppose

that a ∈ α. We have to show that a ∈ Jα. Since α is a strong ideal point,

there is a β ∈ X such that a ∈ β and β ⊆ α. So a ∈ J β ⊆ Jα.

From Proposition 3.2.20 we obtain:

Theorem 3.2.21 ΩB and PtB can be restricted to a pair of functors ΩBS : TopBS →

BFTopS and PtBS : BFTopS → TopBS, and η is the unit of an adjunction ΩBS a

PtBS.

3.3 Three equivalences

The main purpose of formal topology is to obtain a predicative version of the theory of

locales. The connection between formal topology and locale theory is made explicit in

[7], which includes a proof that the category FTop of formal topologies is equivalent

to the category of set-generated Locales, which in turn is equivalent to the category

of locales in the impredicative system IZF.

In this section we shall begin by giving an account of the equivalence between

formal topologies and set-generated Locales. We shall then describe two similar

equivalences, involving OFTop and BFTop respectively. The first of these does not

add much to the result from [7]; however it is hoped that the latter will provide

more insight into the relationship between Sambin’s version of formal topology with

a binary positivity predicate and the more mainstream locale theory.

Before we begin, we shall need the following result of Mac Lane concerning equiv-

alences of categories:

Theorem 3.3.1 If C and D are categories, F : C → D is a full and faithful functor

and there exist functions G and ε• on ObD such that GY ∈ Ob C and εY is an

isomorphism from FGY to Y for all Y ∈ ObD, then C and D are equivalent.


Proof. See [54], pages 93–94. The existence of the functions G and ε• eliminates the

need for the axiom of choice.

3.3.1 Set-generated Locales

A set-generated Frame is a Frame A together with a set G ⊆ A of generators such

that for all a ∈ A,

• G∩ ↓a is a set

• a =∨

(G∩ ↓a)

Let FrmSG be the category of set-generated Frames and Frame homomorphisms,

and let LocSG = FrmopSG, the category of set-generated Locales.

Lemma 3.3.2 We can define a functor F : FTop → LocSG as follows.

• If S is a formal topology, FS is the Frame Sat(S) of subsets U ⊆ S such that

AU = U , ordered by inclusion, with the set of generators Aa | a ∈ S.

• If r : S → S ′, then Fr is the Frame homomorphism Sat(S ′) → Sat(S) defined

by (Fr)(U ′) = Ar−U ′.

Proof. It is straightforward to check that FS is a Frame, Fr is a Frame homomor-

phism and that F preserves identity and composition (see [7]).

Lemma 3.3.3 F is full and faithful.

Proof. Let S and S ′ be formal topologies and let f : Sat(S ′) → Sat(S) be a Frame

homomorphism. Let

rf = 〈a, a′〉 ∈ S × S ′ | a ∈ f(A′a′)

It is easy to prove that rf is a formal topology map and that it is the unique morphism

r : S → S ′ such that Fr = f .


Lemma 3.3.4 Given a Frame A with a generating set G, let GA be the formal

topology S with S = G and a / U iff a ≤∨U . Then εA : a 7→ (↓a) ∩ G is a Frame

isomorphism from A to FGA, with its inverse given by U 7→∨U .

Theorem 3.3.5 FTop is equivalent to LocSG.

Proof. This follows from Theorem 3.3.1 and Lemmas 3.3.2, 3.3.3 and 3.3.4.

3.3.2 Open locales

A similar equivalence can be obtained for OFTop with very little extra effort. Given

a Locale A, let

POS =a ∈ A

∣∣∣ (∀U ∈ Pow(A))[a ≤

∨U → ∃b ∈ U

]and write POS(a) if a ∈ POS. We will say that A is set-open if, for all a ∈ A,

x ∈ a |POS(x) is a set and a ≤∨x ∈ a |POS(x).

Clearly if A is set-open and has a generating set G then G ∩ POS is a set.

Let OLocSG be the full subcategory of LocSG consisting of the set-open set-

generated Locales.

Theorem 3.3.6 OFTop is equivalent to OLocSG.

Proof. We have to prove versions of Lemmas 3.3.2, 3.3.3 and 3.3.4 for open formal

topologies and set-open set-generated Locales.

For Lemma 3.3.2 we need to check that FS is a set-open Locale. It follows from

Proposition 3.2.7 that POS(U) ⇔ PosU for all U ∈ Sat(S). Openness can then be

proved using Positivity for Pos.

The proof of Lemma 3.3.3 can be used with no modification.

For Lemma 3.3.4 note that we can equip GA with the positivity predicate Pos =

G ∩ POS, and this satisfies Monotonicity and Positivity since A is open.


3.3.3 FC-locales

Frames with formal closed subsets

Working in IZF, a completely prime upper section of a frame A is a subset C ⊆ A

such that C = ↑C, and C G U whenever∨U ∈ C.

Classically there is a bijection between the completely prime upper sections of a

frame A and their complements, which are the lower sections of A that are closed

under∨

, which in turn are precisely the principal ideals of A (that is, the subsets of

the form ↓ a for a ∈ A). Thus the completely prime upper sections are, classically,

the subsets of the form A \ (↓ a) for some a ∈ A.

Definition 3.3.7 (IZF) A frame with formal closed subsets (or FC-frame) is a

frame A together with a set C of completely prime upper sections of A, known as the

formal closed sets of A, such that the union of any set of formal closed sets is formal

closed.

If A and A′ are FC-frames, an FC-frame morphism from A to A′ is a frame

homomorphism f : A → A′ such that f−1(U ′) is formal closed for every formal

closed subset U ′ of A′. Let FCFrm be the category of FC-frames and FC-frame

morphisms, and let FCLoc = FCFrmop, the category of locales with formal closed

subsets or FC-locales.

The reader may find it helpful to compare this with the definition of a continu-

ous map between topological spaces. In contrast to the collection of open sets of a

topological space, the formal closed subsets of an FC-frame need not be closed under

finite intersections.

Set-generated frames with formal closed subsets

In order to obtain a definition which makes sense in CZF, we shall use set-generated

Frames and describe the formal closed subclasses in terms of the generators.


Lemma 3.3.8 (IZF) If A is a frame with a set G of generators and C is a completely

prime upper section of A, then C = ↑(C∩G). Therefore every completely prime upper

section is uniquely determined by its intersection with G.

Proof. If a ∈ C then a =∨U for some U ⊆ G. There is a b ∈ U ∩C since

∨U ∈ C.

b ≤ a, so a ∈↑(C ∩G). Conversely if a ∈↑(C ∩G) then a ∈↑C ⊆ C.

We are now in a position to adapt the definition of an FC-frame to CZF.

Definition 3.3.9 A set-generated Frame with formal closed subclasses (or a

set-generated FC-Frame) is a Frame A with a generating set G and a subclass

C0 ⊆ Pow(G) such that

(i) the union of any set of elements of C0 is in C0

(ii) if C0 ∈ C0, V ⊆ G and∨V ∈↑C0 then C0 G V .

We shall call C0 the FC-base of A.

Given C0 ∈ C0, it follows from (ii) that C0 = (↑C0) ∩ G and that if V ⊆ A and∨V ∈↑C0 then ↑C0 G V .

We shall say that a subclass C ⊆ A is formal closed if C = ↑C0 for some C0 ∈ C0,

or equivalently (by the proof of Lemma 3.3.8) if C ∩G ∈ C0.

If A,A′ are set-generated FC-Frames with FC-bases C0 ⊆ Pow(G) and C ′0 ⊆

Pow(G′) respectively, an FC-Frame morphism from A to A′ is a Frame homomor-

phism such that the preimage of any formal closed subclass of A′ is formal closed in

A. Let FCFrmSG be the category of set-generated FC-Frames and FC-Frame mor-

phisms, and let FCLocSG = FCFrmopSG, the category of set-generated FC-Locales.

Lemma 3.3.10 If f : A → A′ is a Frame homomorphism between set-generated

frames and C ′0 ⊆ G′ satisfies part (ii) of Definition 3.3.9, then f−1(↑C ′

0) = ↑f−1(C ′0).

Proof. Suppose that a ∈ f−1(↑ C ′0). Then a =

∨b ∈ G | b ≤ a, so f(a) =∨

f(b) | b ∈ G& b ≤ a ∈ ↑ C ′0. There must be a b ∈ G such that b ≤ a and

f(b) ∈↑C ′0, so a ∈↑f−1(C ′

0).


Conversely suppose that a ∈ ↑f−1(C ′0). There is a b ≤ a such that f(b) ∈ C ′

0.

f(b) ≤ f(a) so f(a) ∈↑C ′0 and hence a ∈ f−1(↑C ′

0).

Lemma 3.3.11 If A and A′ are set-generated FC-Frames with FC-bases C0 and C ′0respectively and f : A → A′ is a Frame homomorphism, then the following are

equivalent:

(i) f is an FC-Frame morphism A→ A′

(ii) (∀C ′0 ∈ C ′0)(∃C0 ∈ C0)[↑C0 = f−1(↑C ′

0)]

(iii) (∀C ′0 ∈ C ′0)[f−1(↑C ′

0) ∩G ∈ C0]

(iv) (∀C ′0 ∈ C ′0)(∃C0 ∈ C0)[C0 ⊆ f−1(↑C ′

0) & f−1(C ′0) ⊆↑C0]

Proof. (i) ⇔ (ii) is immediate from the definitions. (ii) ⇔ (iii) follows from the

remarks following Definition 3.3.9 and (ii) ⇔ (iv) follows from Lemma 3.3.10.

It is straightforward to verify that in IZF the categories FCLoc and FCLocSG

are equivalent. Starting with a locale A with a generating set G and an FC-base

C0 ⊆ Pow(G), we obtain an object of FCLoc by taking C = ↑ C0 |C0 ∈ C0.

Conversely starting with an FC-locale A with a set C of formal closed subsets we may

take the generating set G = A and the FC-base C0 = C.

Equivalence with the category of balanced formal topologies

Lemma 3.3.12 We can define a functor F : BFTop → FCLocSG as follows.

• If S is a balanced formal topology, FS is the Frame Sat(S) of subsets U ⊆

S such that AU = U , ordered by inclusion, with the set of generators G =

Aa | a ∈ S and the FC-base

C0 = Aa | a ∈ U | U ∈ Pow(S) & U = JU

• If r : S → S ′, then Fr is the Frame homomorphism Sat(S ′) → Sat(S) defined

by (Fr)(U ′) = Ar−U ′.


Proof. From the proof of Lemma 3.3.2, it remains to prove that C0 satisfies the

conditions of Definition 3.3.9, and that the Frame homomorphism Fr satisfies one of

the equivalent conditions of Lemma 3.3.11.

To show that C0 is closed under unions, it suffices to prove that the class U ∈

Pow(S) | JU = U is closed under unions. Suppose that JUi = Ui for all i ∈ I, and

let U =⋃

i∈I Ui. If a ∈ U then a ∈ Ui for some i ∈ I, and hence a ∈ JUi ⊆ JU .

Thus JU = U .

Now suppose that C0 ∈ C0 and D ⊆ G, so C0 = Aa | a ∈ V for some

V ⊆ S with J V = V , and D = Aa | a ∈ U for some U ⊆ S. Suppose that∨D ∈↑C0, that is, there is a b ∈ V such that Ab ⊆ AU . Then b ∈ AU ∩ J V so

by Compatibility there is a c ∈ U ∩ J V , so Ac ∈ C0 ∩D.

Given a balanced formal topology map r : S → S ′, we will show that Fr satisfies

part (iv) of Lemma 3.3.11. Suppose that C0 ∈ C0, i.e. C0 = Aa | a ∈ U for some

U ⊆ S such that U = JU . Let C ′0 = A′a′ | a′ ∈ U ′, where U ′ = rU . We have

U ′ = JU ′ by FTM3′ and Proposition 3.2.15. We need to show that C ′0 ⊆ (Fr)−1(↑C0)

and (Fr)−1(C0) ⊆↑C ′0.

For the first inclusion, suppose that a′ ∈ U ′, so ara′ for some a ∈ U . Then

(Fr)(A′a′) = Ar−A′a′ = Ar−a′ ⊇ Aa ∈ C0, so (Fr)(A′a′) ∈↑C0.

For the second inclusion, suppose that V ′ ∈ (Fr)−1(C0). Then V ′ = A′V ′, and

there is an a ∈ U such that (Fr)(V ′) = Aa, that is, Ar−V ′ = Aa. a ∈

Ar−V ′ ∩JU , so by Compatibility there is a b ∈ r−V ′ ∩JU . There is a b′ ∈ V ′ such

that brb′, and hence b′ ∈ rU = U ′. A′b′ ⊆ V ′ and A′b′ ∈ C ′0, so V ′ ∈↑C ′

0.

Lemma 3.3.13 F is full and faithful.

Proof. From Lemma 3.3.3 it remains to prove that rf : S → S ′ satisfies FTM3′

whenever S and S ′ are balanced formal topologies and f : Sat(S ′) → Sat(S) is an

FC-Frame morphism. Suppose that anr∗fU ′ and arfa′. We have to show that a′n′U ′.

Let C0 = Ab | b ∈ J r∗fU ′. There is a C ′0 ∈ C ′0 such that f−1(↑C0) ∩G′ = C ′

0;

C ′0 = A′b′ | b′ ∈ V ′ for some V ′ ∈ Pow(S) such that V ′ = J ′V ′.


Given b′ ∈ V ′, f(A′b′) ∈↑ C0, so there is a b ∈ J r∗fU ′ such that Ab ⊆

f(A′b′). brfb′ and b ∈ r∗fU ′, so b′ ∈ U ′. Thus V ′ ⊆ U ′, so V ′ ⊆ JU ′.

From arfa′ we obtain f(A′a′) ⊇ Aa ∈ C0, so A′a′ ∈ C ′

0. It follows that

A′a′ G V ′, so by Compatibility a′ ∈ V ′ ⊆ JU ′.

Lemma 3.3.14 Given a set-generated FC-Frame A with generating set G and FC-

base C0 ⊆ Pow(G), let GA be the balanced formal topology S with S = G, a / U iff

a ≤∨U and a n U iff (∃C0 ∈ C0)[a ∈ C0 ⊆ U ]. Then εA : a 7→ (↓ a) ∩ G is an

isomorphism from A to FGA in FCFrmSG, and its inverse is ε′A : U 7→∨U .

Proof. From Lemma 3.3.4 it remains to prove that the Frame homomorphisms εA

and ε′A are morphisms of FCFrmSG.

FGA is the frame of subsets U ⊆ G such that U =↓(∨U), with the FC-base

C ′0 = (↓a) ∩G | a ∈ U |U = JU

= (↓a) ∩G | a ∈ C0 |C0 ∈ C0

Given C0 ∈ C0,

ε−1A ((↓a) ∩G | a ∈ C0) = a ∈ G | (∃b ∈ C0)[(↓a) ∩G = (↓ b) ∩G] = C0

(ε′A)−1(C0) = U ⊆ G |U = (↓∨U) ∩G &

∨U ∈ C0 = (↓ a) ∩G | a ∈ C0

So εA and ε′A are morphisms of FCFrmSG

Theorem 3.3.15 The categories BFTop and FCLocSG are equivalent in CZF.

Proof. This follows from Theorem 3.3.1 and Lemmas 3.3.12, 3.3.13 and 3.3.14.

3.3.4 Constructing richer bases

One of the features of Chapter II of [33] is a proof that, given a formal topology

S and a finite family of finitary operations on the opens of S, we can construct a

formal topology isomorphic to S whose base is closed under these operations. The


equivalence between the categories FTop and LocSG can be used to obtain a simple

proof of this result for more general families of operations.

Given a formal topology S, suppose that S ′ is a set of generators for Sat(S), that

is, S ′ ∈ Pow(Pow(S)) and

(∀U ∈ Pow(S))[U /⋃V ∈ S ′ |V / U]

(Note that V ∈ S ′ |V / U is always a set by Restricted Separation.) Let A be the

set-generated Locale FS, where F is the functor defined in Lemma 3.3.2, and let B

be the Locale Sat(S) with the generating set S ′. A and B have the same underlying

Locale, so they are isomorphic in LocSG.

Now let S ′ be the formal topology GB, which has the underlying set S ′. Then

S ∼= GFS = GA ∼= GB = S ′.

A similar construction can be carried out for open formal topologies and balanced

formal topologies. In the case of open formal topologies, we need to check that

POS ∩ S ′ is a set, where POS is the class of positive elements of Sat(S). But

POS ∩ S ′ = U ∈ S ′ |U G Pos

where Pos is the positivity predicate of S, and this is a set by Restricted Separation.

For balanced formal topologies, if C0 is the FC-base for the set-generated FC-

Locale FS, we need to give the Locale B an FC-base C ′0 such that C0 and C ′0 give rise

to the same family of formal closed subclasses of Sat(S). This can be done by taking

C ′0 = (↑C0) ∩ S ′ |C0 ∈ C0

= V ∈ S ′ | (∃U ∈ C0)[U / V ] | C0 ∈ C0

If S is a formal topology (possibly open or balanced) and (fi)i∈I is a set-indexed

family of finitary Functions fi : (Sat(S))ni → Sat(S), we can inductively define the

smallest subclass S ′ of Sat(S) which is closed under these operations and contains

Aa for all a ∈ S. Assuming REA1, S ′ is a set by Theorem 1.1.4.

1REA is not needed if each fi is unary.


Thus given any formal topology S with a set-indexed family of finitary operations

as described above, we can find another formal topology S ′ isomorphic to S whose

base is closed under these operations. This construction is useful if we need to assume

that the base of a formal topology has, for example, binary meets and joins.

3.4 Quasi-formal topologies

In Section 3.1 we defined a formal topology to be a preordered set S together with a

Relation / ⊆ S × Pow(S) such that AU = a ∈ S | a / U is a set for each subset U

of S, and A satisfies the axioms A1-A3. In this section we shall consider the effect

of dropping the condition that AU is a set. We shall not pursue this idea very far in

this thesis because it differs from the usual approach to formal topology, but we shall

see in Chapter 4 that our more general approach does have some potential benefits.

In particular, it is possible to inductively generate quasi-formal topologies in CZF

without using the Regular Extension Axiom, and furthermore every quasi-formal

topology can be inductively generated by a class-covering system (see Section 4.4).

One question that we have left unanswered is whether there is an equivalence

between the category of quasi-formal topologies and some category similar to that of

(set-generated) Frames. Two candidates for the corresponding kind of “Frame” are

as follows.

(i) The elements of the Frame are classes. There is a set-indexed family of gen-

erators such that every element of the Frame is the join of some “family” of

generators, which need not be a set.

(ii) The elements of the Frame are sets, and the Frame itself is a class. There is a

set of generators such that every element of the Frame is a join of some set of

generators, but the class of all generators below a given element of the Frame is

not required to be a set. The condition that a ≤ b & b ≤ a⇒ a = b is omitted

from the definition of a Frame. Only joins of subsets of the frame need exist.

The two different definitions can be illustrated by considering the open subclasses


of a ct-space X . The collection of open subclasses of X forms a Frame of type (i),

isomorphic to the saturated subclasses of S (that is, those of the form Y ). This

approach has the drawback that it is difficult to state exactly what we mean by a

“Frame” in this context: the elements of the frame do not form a class-indexed family

of classes, so we would need to find some more general notion of a family of classes.

A Frame of type (ii) can be obtained by preordering Pow(S) by U ≤ V iff extU ⊆

extV . This kind of Frame is easier to formalise in CZF, and our definition of quasi-

formal topologies was formulated with this in mind. This approach also has some

problems; in particular an extra condition needs to be added to the definitions of

ct-spaces and quasi-formal topologies to ensure that the corresponding Frames have

binary meets.

3.4.1 The category of quasi-formal topologies

Definition 3.4.1 A quasi-formal topology is a set S with a Preorder ≤ and a Rela-

tion / ⊆ S × Pow(S) such that

A1: ↓U ⊆ AU

A2: U ⊆ AV ⇒ AU ⊆ AV

A3: AU ∩ AV ⊆ A+(U ↓V )

for all sets U, V ∈ Pow(S), where

AU = a ∈ S | a / U

A+W =⋃AW ′ |W ′ ∈ Pow(W )

for each subset U of S and each subclass W of S.

It is easy to see that if U is a subset of S then AU = A+U , provided A1 and A2

hold. As with ordinary formal topologies, we shall use S, T , . . . to denote quasi-formal

topologies.

Lemma 3.4.2 If S is a quasi-formal topology, U ∈ Pow(S) and U ⊆ A+V for some

subclass V ⊆ S, then there is a subset V ′ ⊆ V such that U ⊆ AV ′.


Proof. From U ⊆ A+V we have

(∀a ∈ U)(∃W )[W ⊆ V & a / W ]

By Strong Collection there is a set V such that

(∀a ∈ U)(∃W ∈ V)[W ⊆ V & a / W ]

and (∀W ∈ V)(∃a ∈ U)[W ⊆ V & a / W ]

Let V ′ =⋃V . Then V ′ ⊆ V and U ⊆ AV ′.

Proposition 3.4.3 If S is a quasi-formal topology then the following hold for any

subclasses U, V ⊆ S.

A1+: ↓U ⊆ A+U

A2+: U ⊆ A+V ⇒ A+U ⊆ A+V

A3+: A+U ∩ A+V ⊆ A+(U ↓V )

Proof. A1+: If a ≤ b ∈ U then a ∈ Ab ⊆ A+U .

A2+: Suppose that U ⊆ A+V and a ∈ A+U . There is a subset U ′ ⊆ U such that

a ∈ AU ′, and by Lemma 3.4.2 there is a subset V ′ ⊆ V such that U ′ ⊆ AV ′.

Thus a ∈ AV ′ ⊆ A+V by A2.

A3+: Suppose that a ∈ A+U ∩ A+V . There exist U ′ ∈ Pow(U) and V ′ ∈ Pow(V )

such that a ∈ AU ′ ∩ AV ′. By A3, a ∈ A+(U ′ ↓ V ′), so there is a subset

W ′ ⊆ U ′ ↓V ′ such that a / W ′. U ′ ↓V ′ ⊆ U ↓V , so a ∈ A+(U ↓V ).

The need for Strong Collection in the proof of A2+ seems rather surprising, but it

appears to be necessary. Our proof is very similar to Gambino’s proof in [38, §5.2.1]

that a nucleus on the lower sets of a poset can be “lifted” to a nucleus on the lower

classes of that poset, the only significant difference being that here AU may be a

proper class rather than a set.


Although Definition 3.4.1 appears to be the right one to use when considering

inductively generated quasi-formal topologies (see Section 4.4), it seems slightly too

general when it comes to considering the relationship with Frames, and with ct-spaces.

The problem is that the collection of classes of the form AU (where U ∈ Pow(S))

does not appear to have finite meets. To solve this, we shall say that a quasi-formal

topology S is standard if

(∀a, b ∈ S)(∃W )[W ⊆ a↓b ⊆ AW ]

Proposition 3.4.4 If S is a standard quasi-formal topology and U, V ∈ Pow(S),

then there is a W ∈ Pow(U ↓V ) such that AU ∩ AV = AW .

Proof. Given U, V ∈ Pow(S) we have

(∀p ∈ U × V )(∃W )(∃a, b ∈ S)[p = 〈a, b〉 & W ⊆ a↓b ⊆ AW ]

so by Strong Collection there is a set W such that

(∀p ∈ U × V )(∃W ∈ W)(∃a, b ∈ S)[p = 〈a, b〉 & W ⊆ a↓b ⊆ AW ]

and (∀W ∈ W)(∃p ∈ U × V )(∃a, b ∈ S)[p = 〈a, b〉 & W ⊆ a↓b ⊆ AW ]

Let W ′ =⋃W . Then W ′ ⊆ U ↓V and U ↓V ⊆ AW ′.

Definition 3.4.5 If S and S ′ are quasi-formal topologies, a quasi-formal topology

map S → S ′ is a subclass r ⊆ S × S ′ such that

QFTM1: S ⊆ A+r−S ′

QFTM2: r−a′ ∩ r−b′ ⊆ A+r−(a′ ↓b′) for all a′, b′ ∈ S ′

QFTM3: r−A′U ′ ⊆ A+r−U ′ for all U ′ ∈ Pow(S ′)

QFTM4: (∀a′ ∈ S ′)(∃U ∈ Pow(S))[r−a′ = AU ]

Lemma 3.4.6 If r : S → S ′ is a quasi-formal topology map, then r−U ′ ↓ r−V ′ ⊆

A+r−(U ′ ↓V ′) for all subclasses U ′, V ′ ⊆ S ′.


Proof. Suppose that c ∈ r−U ′ ↓ r−V ′, so there exist a′ ∈ U ′ and b′ ∈ V ′ such that

c ∈↓ r−b′ ⊆ r−b′ and c ∈↓ r−c′ ⊆ r−c′. By QFTM2, c ∈ A+r−(a′ ↓ b′) ⊆

A+r−(U ′ ↓V ′).

Lemma 3.4.7 If r : S → S ′ is a quasi-formal topology map, then r−A′+U ′ ⊆

A+r−U ′ for every subclass U ′ ⊆ S ′.

Proof. If a ∈ r−A′+U ′, then ara′ for some a′ ∈ A′+U ′. There is a subset V ′ ⊆ U ′

such that a′ /′ V ′, so a ∈ r−A′V ′ ⊆ A+r−V ′ ⊆ A+r−U ′.

Proposition 3.4.8 The quasi-formal topologies and quasi-formal topology maps form

a category, QFTop.

Proof. As with FTop, the identity map on a quasi-formal topology S is given by

idS = 〈a, b〉 ∈ S × S | a ∈ Ab

and given r : S → S ′ and r′ : S ′ → S ′′, their composite is

r′ r = 〈a, a′′〉 ∈ S × S ′′ | a ∈ A+r−r′−a′′

We need to check that r′ r is a quasi-formal topology map.

QFTM1: By QFTM3 for r′, S ′ ⊆ A′+r′−S ′′, so using Lemma 3.4.7 for r we have

r−S ′ ⊆ A+r−r′−S ′′ ⊆ A+((r′ r)−S ′′). QFTM1 for r gives S ⊆ A+r−S ′, so

S ⊆ A+((r′ r)−S ′′).

QFTM2: Let a′′, b′′ ∈ S ′′. Using Lemmas 3.4.6 and 3.4.7 for r and r′, we obtain

A+r−r′−a′′ ∩ A+r−r′−b′′ ⊆ A+(r−r′−a′′↓r−r′−b′′)

⊆ A+r−(r′−a′′↓r′−b′′)

⊆ A+r−A+r′−(a′′ ↓b′′)

⊆ A+r−r′−(a′′ ↓b′′)

⊆ A+((r′ r−)(a′′ ↓b′′))


QFTM3: Given U ′′ ∈ Pow(S ′′), (r′ r)−A′′U ′′ ⊆ A+r−r′−A′′U ′′ ⊆ A+r−A′+r′−U ′′

by QFTM3 for r′. Lemma 3.4.7 for r gives r−A′+r′−U ′′ ⊆ A+r−r′−U ′′ ⊆

A+((r′ r)−U ′′), so (r′ r)−A′′U ′′ ⊆ A+((r′ r)−U ′′).

QFTM4: Given a′′ ∈ S ′′, by QFTM4 for r′ we have r′−a′′ = AU ′ for some U ′ ∈

Pow(S ′). QFTM4 for r gives

(∀a′ ∈ S ′)(∃U)[U ⊆ S & r−a′ = AU ]

Applying Strong Collection, there is a set U such that

(∀a′ ∈ U ′)(∃U ∈ U)[U ⊆ S & r−a′ = AU ]

and (∀U ∈ U)(∃a′ ∈ U ′)[U ⊆ S & r−a′ = AU ]

Let V =⋃U . Then AV = A+r−U ′ = A+r−AU ′ = A+r−r′−a′′.

Let QFTopS be the full subcategory of QFTop whose objects are the standard

quasi-formal topologies.

3.4.2 A Galois Adjunction

Quasi-standard ct-spaces

For ct-spaces, we need a notion of “quasi-standardness” similar to that of standard-

ness for quasi-formal topologies. We will say that a ct-space X = 〈X, , S〉 is quasi-

standard if

(∀a, b ∈ S)(∃W ∈ Pow(S))[extW = ext a ∩ ext b]

Note that every standard ct-space is quasi-standard. If X is quasi-standard, we can

easily show that

(∀U, V ∈ Pow(S))(∃W ∈ Pow(S))[extW = extU ∩ extV ]

The proof is similar to that of Proposition 3.4.4.


Proposition 3.4.9 Given a quasi-standard ct-space X = 〈X, , S〉, define a Pre-

order on S by a ≤ b iff ext a ⊆ ext b, and a Relation / ⊆ S × Pow(S) by taking a / U

iff ext a ⊆ extU .

S, ≤ and / form a standard quasi-formal topology ΩQSX .

Proof. It is easy to check that / satisfies A1 and A2. Given a, b ∈ S, there is a

W ∈ Pow(S) such that ext a ∩ ext b = extW , and it follows that W ⊆ a↓b ⊆ AW .

From the proof of Proposition 3.4.4 we can see that for any U, V ∈ Pow(S) there

is a W ∈ Pow(U ↓V ) such that AU ∩ AV = AW ⊆ A+(U ↓V ).

If X and X ′ are quasi-standard ct-spaces, a continuous map f : X → X ′ is

quasi-standard continuous if for all a′ ∈ S ′ there is a set U ∈ Pow(S) such that

f−1(ext a′) = extU .

Proposition 3.4.10 The quasi-standard ct-spaces and quasi-standard continuous

maps form a subcategory TopQS of Top.

Proof. Clearly the identity map on a ct-space is quasi-standard continuous. We need

to show that the composite of two quasi-standard continuous maps is quasi-standard

continuous, so suppose that f : X → X ′ and f ′ : X ′ → X ′′ are quasi-standard

continuous.

Let a′′ ∈ S ′′. Since f ′ is quasi-standard continuous, there is a subset U ′ ⊆ S ′ such

that f−1(ext a′′) = extU ′. Since f is quasi-standard continuous, we have

(∀a′ ∈ U ′)(∃V )[V ⊆ S & f−1(ext a′) = extV ]

so by Strong Collection there is a set V such that

(∀a′ ∈ U ′)(∃V ∈ V)[V ⊆ S & f−1(ext a′) = extV ]

and (∀V ∈ V)(∃a′ ∈ U ′)[V ⊆ S & f−1(ext a′) = extV ]

Let U =⋃V . Then extU = f−1(extU ′) = (f ′ f)−1(extU ′′).


From quasi-formal topologies to ct-spaces

Definition 3.4.11 If S is a quasi-formal topology, a formal point of S is a subset

α ⊆ S satisfying the rules FP1-FP3 of Definition 3.2.1.

As with ordinary formal topologies, the formal points of a quasi-formal topology

S form a ct-space PtQ(S) = 〈Pt(S),3, S〉, where Pt(S) is the class of formal points

of S. PtQ can be extended to a functor PtQ : QFTop → Top; for each quasi-formal

topology map r : S → S ′, define PtQ(r) : PtQ(S) → PtQ(S ′) by

PtQ(r)(α) = rα = a′ ∈ S ′ | (∃a ∈ α)[ara′]

for each α ∈ PtQ(S).

Proposition 3.4.12 Given a quasi-formal topology map r : S → S ′, rα is a set for

each formal point α of S.

Proof. Given a′ ∈ S ′, by QFTM4 there is a U ∈ Pow(S) such that r−a′ = AU .

From FP3 we have r−a′ G α if and only if U G α, so

0 | r−a G α = 0 |U G α

and this is a set by Restricted Separation. Hence

rα = a′ ∈ S ′ | 0 ∈ 0 | r−a′ G α

is a set by Restricted Separation.

Having shown that rα is a set, it is straightforward to check that it satisfies the

axioms FP1-FP3.

Proposition 3.4.13 PtQ can be restricted to a functor PtQS : QFTopS → TopQS.

Proof. If S is a standard quasi-formal topology and a, b ∈ S, then there is a W ∈

Pow(S) such thatW ⊆ a↓b ⊆ AW , and it follows by FP2 and FP3 that ext a∩ext b =

extW .


Given a quasi-formal topology map r : S → S ′ between standard quasi-formal

topologies, let X = PtQ(S), X ′ = PtQ(S ′) and f = PtQ(r). Given a′ ∈ S ′, by

QFTM4 there is a U ∈ Pow(S) such that r−a′ = AU . For each α ∈ PtQ(S),

α ∈ f−1(ext′ a′) ⇔ a′ ∈ rα⇔ α G r−a′ ⇔ α G AU ⇔ α G U

so f−1(ext′ a′) = extU . Thus f is quasi-standard continuous.

The adjunction

The unit of the Galois adjunction will, again, be given by the continuous maps

ηX : X → Sob(X ) for each ct-space X , where ηX (x) = αx for all x ∈ X.

Theorem 3.4.14 η is the unit of a Galois adjunction ΩQS a PtQS.

This is a consequence of the following Proposition:

Proposition 3.4.15 Let X = 〈X, , S〉 be a quasi-standard ct-space.

(i) ΩQSX is spatial (as defined in Definition 3.2.2).

(ii) PtQS(ΩQSX ) = Sob(X ) and ηX is quasi-standard continuous.

(iii) If S ′ is a standard quasi-formal topology and g : X → PtQS(S ′) is quasi-standard

continuous, then


is the unique quasi-formal topology map r : ΩQSX → S ′ such that g = PtQS(r).

(iv) If X = PtQS(S) for some quasi-formal topology S, then X is sober.

Proof. The proof is similar to that of Proposition 3.2.5.

(i) can be proved without any modification.

(ii) We have to check that ηX is quasi-standard continuous. Let X ′ = Sob(X ). Then

given a ∈ S, η−1X (ext′ a) = x ∈ X | a ∈ αx = exta.


(iii) We have to show that rg is a quasi-formal topology map.

QFTM1: Since g is quasi-standard continuous, we have

(∀a′ ∈ S ′)(∃U)[U ⊆ S & g−1(ext′ a) = extU ]

so by Strong Collection there is a set U such that

(∀a′ ∈ S ′)(∃U ∈ U)[U ⊆ S & g−1(ext′ a) = extU ]

and (∀U ∈ U)(∃a′ ∈ S ′)[U ⊆ S & g−1(ext′ a) = extU ]

Let U =⋃U . Then U ⊆ r−g S

′, and extU = g−1(ext′ S ′) = g−1(X ′) = X,

so S ⊆ AU ⊆ A+r−g S′.

QFTM2: Let a′, b′ ∈ S ′. Since S ′ is standard, there is aW ′ ∈ Pow(S ′) such that

W ′ ⊆ a′ ↓ b′ ⊆ A′W ′. By QFTM4 (proved below), (∀c′ ∈ W ′)(∃U)[U ⊆

S & r−g c′ = AW ]. Making a straightforward application of Strong

Collection we can obtain a set W ∈ Pow(S) such that W ⊆ r−g W′ ⊆ AW .

If c ∈ r−g a′ ∩ r−g b′ then

ext c ⊆ g−1(ext a′) ∩ g−1(ext b′) = g−1(extW ′) = ext r−g W′ = extW

Thus r−g a′ ∩ r−g b′ ⊆ AW ⊆ A+r−g W′ ⊆ A+r−g (a′ ↓b′).

QFTM3: Suppose that a ∈ r−g A′U ′, that is, arga′ for some a′ ∈ A′U ′. Then

ext a ⊆ g−1(ext′ a′) ⊆ g−1(ext′ U ′). Since g is quasi-standard continuous

we have (∀b′ ∈ U ′)(∃U)[U ⊆ S & g−1(ext′ b′) = extU ], so making use of

Strong Collection we can construct a set U ∈ Pow(S) such that (∀b ∈

U)(∃b′ ∈ U ′)[ext b ⊆ g−1(ext′ b′)] and g−1(ext′ U ′) ⊆ extU . Hence a ∈ AU

and U ⊆ r−g U′, so a ∈ A+r−g U

′.

QFTM4: Let a′ ∈ S ′. Since g is quasi-standard continuous there is a set

U ∈ Pow(S) such that extU = g−1(ext a′), and it follows from this that

r−g a′ = AU .

This completes the proof that rg is a quasi-formal topology map. The fact that

it is the unique quasi-formal topology map r such that g = Pt(r) can be proved

in the same way as for Proposition 3.2.5.


(iv) can be proved as in Proposition 3.2.5.

3.5 Subspaces

The notion of a subspace of a formal topology is analogous to that of a subset of a

topological space, with the induced topology.

In locale theory, a sublocale inclusion is a regular monomorphism A′ → A. A

locale morphism A′ → A is a sublocale inclusion if and only if the corresponding

frame homomorphism A→ A′ is surjective (see [45]). One sublocale inclusion A′ → A

is smaller than another A′′ → A if the former inclusion factors (uniquely) through

the latter. A sublocale of A is an equivalence class of sublocale inclusions into A,

where two sublocale inclusions are regarded as “equal” if each factors through the

other.

Each sublocale of A can be represented by a canonical sublocale inclusion, in

which the domain is the locale of fixed points of a nucleus on A (again, see [45]).

There is precisely one nucleus on A for every sublocale, and conversely every nucleus

gives rise to a sublocale of A.

In this section we describe the predicative version of sublocale inclusions and

nuclei, working in formal topology. The definition of a subspace of a formal topology

can be found in [33], together with some examples of subspaces, including the image

of a formal topology map and open, closed and dense subspaces. We shall begin

by revisiting these notions and proving a few additional facts about them, and then

consider the problem of defining weakly closed and strongly dense subspaces in a

predicative way.

3.5.1 Subspaces of formal topologies

Subspace inclusions

The predicative version of sublocales has been considered by Curi in [33]. A formal

topology map r : S ′ → S is a subspace inclusion iff the map U 7→ A′r−U is a surjection


from Sat(S) onto Sat(S ′), that is, for all U ′ ∈ Pow(S ′) there is a U ∈ Pow(S) such

that A′U ′ = A′r−AU (or equivalently A′U ′ = A′r−U). It can be shown that this is

true if and only if U ′ /′ r−r−∗A′U ′ for all U ′ ∈ Pow(S ′).

A subspace inclusion r1 : S1 → S is smaller than r2 : S2 → S (written r1 v r2) if

and only if there is a formal topology map r : S1 → S2 such that r1 = r2 r. Two

subspace inclusions r1 and r2 are considered to be equal if r1 v r2 and r2 v r1.

Proposition 3.5.1 If r1 v r2 and r2 v r1 then S1 and S2 are isomorphic.

Images of formal topology maps

Given a formal topology map r : S ′ → S, we can define a covering relation /r on S

by setting ArU = r−∗A′r−U for all U ∈ Pow(S). Let Sr be the formal topology on

S with the same preorder ≤ as in S, and the covering relation /r. It is easily verified

that Ar satisfies the axioms A1-A3, and that AU ⊆ ArU for all U ∈ Pow(S). r

satisfies the axioms for a formal topology map S ′ → Sr. We shall call Sr the image

of r.

If S ′ is open, with positivity predicate Pos′, the set

r Pos′ = a ∈ S | Pos′(r−a)

is a positivity predicate for Sr, so Sr is open.

Canonical subspace inclusions

If S is a formal topology, then given any covering relation /′′ on S (with the same

preorder ≤) such that AU ⊆ A′′U for all U ∈ Pow(S), it can be shown that the

relation

i = 〈a, b〉 ∈ S × S | a ∈ A′′b

is a subspace inclusion from S ′′ into S, and S ′′ is the image of i. A subspace inclusion

of this form will be called a canonical subspace inclusion, or just a subspace. In

particular the image of any formal topology map is a canonical subspace inclusion,

and every canonical subspace inclusion is equal to its own image.


Proposition 3.5.2 If r : S ′ → S is a subspace inclusion, Sr is its image and

i : Sr → S is the corresponding canonical subspace inclusion, then r and i repre-

sent equal subspaces of S.

Sketch of proof. The relation r ⊆ S ′ × S can also be interpreted as a formal

topology map r : S ′ → Sr, and

s = 〈a, a′〉 ∈ S × S ′ | a ∈ r−∗A′a′

is a formal topology map s : Sr → S ′ (this is a straightforward proof using the fact

that U ′ / r−s−U ′ for all U ′ ∈ Pow(S ′)). i = r s and r = i r, so i v r and r v i.

If i : S ′ → S is a canonical subspace inclusion, we will write S ′ v S. Given

two such subspaces S ′ and S ′′ of S, S ′ is a smaller subspace than S ′′ if and only if

A′′U ⊆ A′U for all U ∈ Pow(S), that is S ′ v S ′′.

Remark. This approach differs slightly from the one in [33]. In [33], an extra

condition on A′ was needed to ensure that i satisfies FTM2 (that is, preserves finite

meets). This is not necessary here because S and S ′ share the same preorder ≤.

Formal points of subspaces

Given a ct-space X = 〈X, , S〉 and a subset X ′ ⊆ X, we can define a new ct-space

X ′ = 〈X ′, ′, S〉 where ′ is the restriction of to X ′ × S. ΩX ′ is a subspace of the

formal topology ΩX (if it is given the preorder ≤ from ΩX ).

Conversely, starting with a formal topology subspace S ′ v S, it is clear that

Pt(S ′) ⊆ Pt(S) and that Pt(S ′) has the subspace topology.

If S = ΩX and S ′ v S, we obtain a subset X ′ ⊆ X by taking X ′ = x ∈ X |αx ∈

Pt(S ′). For all a ∈ S and U ∈ Pow(S), a /′ U implies ext′ a ⊆ ext′ U ; if the converse

also holds then S ′ = ΩX ′ and so S ′ is spatial.


3.5.2 Open subspaces

Given a subset V ⊆ S, we can obtain a subspace SV of S by taking AVU = a ∈

S | a↓V / U. We shall call a subspace of S an open subspace if it is equal to SV for

some V ∈ Pow(S).

Proposition 3.5.3 (i) S /V U iff V / U .

(ii) SV is the largest subspace S ′ v S such that S /′ V .

Proof. (i) Suppose that S /V U . If a ∈ V then a ∈ a↓V and so a / U .

Now suppose that V / U . Given a ∈ S, a↓V ⊆↓V / U . So S ⊆ AVU .

(ii) Suppose that S ′ v S and S /′ V . If a /V U , then a /′ a↓V /′ U . Thus S ′ v SV .

We also have the following result concerning the points of open subspaces.

Proposition 3.5.4 (i) Pt(SV ) = α ∈ Pt(S) |α G V

(ii) If S = ΩX then SV = ΩXV , where XV = 〈extV, V , S〉 and V is the

restriction of to extV × S.

Proof. (i) If α ∈ Pt(SV ) then there is an a ∈ α and a /V V , so α G V .

Now suppose that α ∈ Pt(S) and α G V . If a ∈ α and a /V U then a ↓V / U .

(a↓V ) G α by FP2, so U G α by FP3. Thus α satisfies FP3 for SV .

(ii) is clear from the definitions.

3.5.3 Closed subspaces and dense maps

Given a subset V ⊆ S, we can obtain a subspace SV of S by taking AVU = A(U∪V ).

The subspaces of this form are known as the closed subspaces of S. Intuitively SV

can be thought of as the complement of the formal open set V .

A formal topology map r : S ′ → S is dense if r−∗A′∅ ⊆ A∅, that is, r−a /′ ∅ ⇒

a / ∅ for all a ∈ S. A dense subspace of S is one in which the subspace inclusion map

is dense; thus S ′ v S is dense iff A′∅ ⊆ A∅.


The closure of a subspace S ′ v S is the closed subspace SZ , where Z = b ∈

S | b /′ ∅. It is observed in [33] that this satisfies the properties expected of a closure

operation, and in particular that the closure of S ′ is the smallest closed subspace of

S containing S ′, and that the closure of a dense subspace of S is equal to S.

Proposition 3.5.5 Pt(SV ) = α ∈ Pt(S) |α ∩ V = ∅

Proof. Suppose that α ∈ Pt(SV ). If a ∈ α ∩ V then a /V ∅, contradicting FP3.

Now suppose that α ∈ Pt(S) and α∩V = ∅. If a ∈ α and a /V U , then a /U ∪V .

By FP3 α G U ∪ V and hence α G U . Thus α satisfies FP3 for SV .

In contrast to open subspaces, a closed subspace of a spatial formal topology need

not be spatial.

Proposition 3.5.6 Let S = a, b, AU = U for all U ∈ Pow(S) and V = b. If

SV is spatial then ¬¬(a = b) → (a = b).

Proof. Suppose that ¬¬(a = b), and let X ′ = Pt(SV ). If α ∈ ext′ a then b /∈ α and

so ¬(a = b); thus ext′ a = ∅ = ext′ ∅. If SV is spatial then a /V ∅ and hence a / V ,

that is a = b.

Corollary 3.5.7 If every closed subspace of a spatial formal topology is spatial then

REM holds.

Proof. Given a restricted formula ϕ, let a = 0 and b = 0 |ϕ. The formal

topology S defined in the previous proposition is spatial, since it arises from the

discrete topology on a, b. a = b↔ ϕ, so if Sb is spatial then ¬¬ϕ→ ϕ.

3.5.4 Weakly closed subspaces and strongly dense maps

The above notion of a closed subspace is the point-free analogue of the complement

of an open set. The definition of the closure of a subset of a topological space from

Section 2.1 is more useful constructively, but harder to formulate in a point-free way.


The definition of density also seems rather negative, as it is the point-free analogue

of the property “if f−1(ext a) is empty then ext a is empty”. In the point-sensitive

case, a far more useful constructive definition of density is “if ext a is inhabited then

f−1(ext a) is inhabited”.

Another approach, which was introduced by Johnstone in [47] in the impredicative

intuitionistic case, is that of weakly closed sublocales and strongly dense sublocales.

This idea was later used in [48] and [86] to obtain “fibrewise” versions of some of the

separation axioms for locales.

In the predicative case, defining “weakly closed”, “weak closure” and “strongly

dense”, and the corresponding separation properties such as weak regularity, seems

to be more difficult than the standard notions of “closed”, “closure” and “dense”

mentioned above, but nevertheless we have been able to make some progress.

Strongly dense subspaces

Definition 3.5.8 A subspace S ′ v S is strongly dense if A′U = AU whenever

U = a ∈ S | 0 ∈ p for some p ∈ Pow(0).

A formal topology map r : S ′ → S is strongly dense if its image is strongly dense

in S.

Lemma 3.5.9 S ′ v S is strongly dense iff a /′ U → a / a | ∃b ∈ U for all a ∈ S

and U ∈ Pow(S).

Proof. Suppose that S ′ v S is strongly dense and a/′U . Let p = 0 | ∃b ∈ U. Then

U ⊆ c ∈ S | 0 ∈ p, so a/′c ∈ S | 0 ∈ p and hence a/a↓c ∈ S | 0 ∈ p/a | 0 ∈ p.

Conversely suppose that a /′ U → a / a | ∃b ∈ U for all U ∈ Pow(S). Let

p ∈ Pow(0) and U = a ∈ S | 0 ∈ p. If a ∈ A′U then a / a | ∃b ∈ U = a | 0 ∈

p ⊆ U ; thus A′U ⊆ AU .

When one of the formal topologies is open, strong density can be characterised

more simply by using the positivity predicate. The following result is a special case

of Lemma 1.11 of [47].


Lemma 3.5.10 If S ′ v S a strongly dense subspace then S is open iff S ′ is open.

In either case, Pos = Pos′.

Proof. Suppose that S is open, and let Pos be its positivity predicate. We will show

that Pos is a positivity predicate for S ′, applying Proposition 3.2.6

Monotonicity: Suppose that Pos a and a/′U . Then a/a | ∃b ∈ U since S ′ is strongly

dense, so ∃b ∈ U by Monotonicity for S.

Positivity: Given a ∈ S, a / a | Pos a and hence a /′ a | Pos a.

Now suppose that S ′ is open, and let Pos′ be its positivity predicate. We will

show that Pos′ is a positivity predicate for S ′.

Monotonicity: If Pos a and a / U , then a /′ U and so Pos′ U by Monotonicity for S ′.

Positivity: Given a ∈ S, let U = a | Pos′ a. a /′ U by Positivity for S ′, so

a / a | ∃b ∈ U = a | Pos′ a.

Proposition 3.5.11 If S ′ v S and S and S ′ are both open, then S ′ is strongly dense

in S if and only if Pos = Pos′.

Proof. The implication from left to right has already been proved in Lemma 3.5.10.

Suppose that Pos = Pos′ and a /′ U . To show that a / a | ∃b ∈ U it suffices to show

that Pos a→ a/a | ∃b ∈ U. But if Pos a then Pos′ a and so ∃b ∈ U by Monotonicity

for S ′.

Corollary 3.5.12 If S ′ v S and S ′ is open, then S ′ is strongly dense in S if and

only if Pos′ satisfies Positivity for S, that is,

(Pos′ a→ a / U) → a / U

Proof. If S ′ is strongly dense then Pos′ satisfies Positivity by Lemma 3.5.10.


Pos′ always satisfies Monotonicity for S since a/U implies a/′U , so if Pos′ satisfies

Positivity for S then it is a positivity predicate for S and hence S ′ is strongly dense

in S by Lemma 3.5.10 and Proposition 3.5.11.

It also follows from Proposition 3.5.11 that if r : S ′ → S is a formal topology map

between open formal topologies, then r is strongly dense if and only if r Pos′ = Pos.

Weak closures and weakly closed subspaces

Definition 3.5.13 If S ′,S ′′ v S, we say that S ′′ is the weak closure of S ′ (in S) iff

S ′′ is the largest subspace of S such that S ′ is a strongly dense subspace of S ′′.

If S ′ is its own weak closure in S, we say that S ′ is weakly closed (in S).

This definition is rather unsatisfactory as it involves quantification over classes,

so “S ′′ is the weak closure of S ′” is not expressed as a formula of CZF. Also we have

no way of constructing the weak closure of a subspace in general, in contrast to the

impredicative case where the weak closure always exists. However, we shall see in

Subsection 4.2.6 that if S ′ v S, S is inductively generated and S ′ is an open formal

topology, then the weak closure of S ′ does exist.

Proposition 3.5.14 A closed subspace of a formal topology is weakly closed.

Proof. Let SV v S be a closed subspace. Suppose that SV v S ′ v S and the

subspace inclusion SV v S ′ is strongly dense. Then SV is dense in S ′, so SV and S ′

are equal.

3.6 Separation properties and compactness

3.6.1 Constructive notions of regularity

The point-free notion of regularity depends on understanding what it means for one

formal open to be “well-inside” another. In the classical point-sensitive case, an open

set Y1 is “well-inside” Y2 if cl(Y1) ⊆ Y2.


The standard point-free approach is obtained by considering the complement of

the closed set Y1, and this gives rise to the notions of “3-well inside” and “3-regularity”

defined below. However, this seems to be too strong constructively for some purposes;

in particular uniform formal topologies cannot be proved to be 3-regular without REM

(see the paragraph following Lemma 5.4.7).

Another approach introduced by Johnstone in [48] for constructive locale theory

is to say that b is well-inside a if the weak closure of b is contained in a (considering

a and b as open sublocales). This presents a problem in CZF because quantification

over classes appears to be needed to express “b well-inside a”, but if S is an open

inductively generated formal topology then the weak closure of b can be constructed as

another inductively generated formal topology, allowing us to define weak regularity

(see Proposition 4.2.7). The definition obtained in this way makes sense even if S

is not inductively generated, and this gives rise to the notions of “1-well inside” and

“1-regularity” defined below.

We shall also consider a new form of regularity, called “2-regularity” below, which

is weaker than 3-regularity and implies 1-regularity. This definition seems to be easier

to work with than 1-regularity, and we shall see that every uniform formal topology

is 2-regular. Although 2-regularity is constructively different from 3-regularity, it is

not yet clear whether 1- and 2-regularity are equivalent.

Definition 3.6.1 Given a formal topology S and a, b ∈ S, b is 1-well inside a if for

all U ∈ Pow(S), if

• a ∈ U

• AU = U

• (∀c ∈ S)[ [Pos(b↓c) → c ∈ U ] → c ∈ U ]

then U = S.

Definition 3.6.2 Given a formal topology S and a, b ∈ S, b is 2-well inside a if

S / c ∈ S | Pos(b↓c) → c / a


Definition 3.6.3 Given a formal topology S and a, b ∈ S, b is 3-well inside a if

S / a ∪ b∗, where

b∗ = c ∈ S | b↓c / ∅

Definition 3.6.4 For n = 1, 2, 3, a formal topology S is n-regular if for all a ∈ S

there is a set W ∈ Pow(S) such that a / W and b is n-well inside a for all b ∈ W .

Proposition 3.6.5 (i) If S is 2-regular then it is 1-regular.

(ii) If S is 3-regular then it is 2-regular.

Proof. (i) We shall show that if b is 2-well inside a then it is 1-well inside a.

Suppose that b is 2-well inside a, and let U ∈ Pow(S) satisfy the conditions of

Definition 3.6.1. Let

V = c ∈ S | Pos(b↓c) → c / a

Then S /V , so to prove that S /U it suffices to show that V ⊆ U . But if c ∈ V

then Pos(b ↓ c) → c / a and Aa ⊆ U , and hence Pos(b ↓ c) → c ∈ U , from

which it follows that c ∈ U .

(ii) We shall show that if b is 3-well inside a then it is 2-well inside a. Suppose that

b is 3-well inside a, so S / a ∪ b∗. Let

U = c ∈ S | Pos(b↓c) → c / a

Clearly a ∈ U . If c ∈ b∗ then ¬Pos(b↓c) and hence c ∈ U . Thus a ∪ b∗ ⊆ U ,

so S / U .

We have the following relationship between R] ct-spaces and 3-regular formal

topologies:

Proposition 3.6.6 (i) If X is a standard ct-space, then X is R] if and only if ΩX

is 3-regular.

(ii) If S is a 3-regular standard formal topology then Pt(S) is R].


Proof. See [8].

For 1- and 2-regularity there does not appear to be such a strong connection with

the point-sensitive notions of regularity from Section 2.2. We do, however, have the

following:

Proposition 3.6.7 If X is a standard ct-space and ΩX is 1-regular then X is R.

Proof. Given x ∈ X and a ∈ αx, we can choose a b ∈ αx such that b is 1-well inside

a in ΩX . We will show that cl(ext b) ⊆ ext a. Let

Y = y ∈ X | y ∈ cl(ext b) → y ∈ ext a

and U = Y . Clearly ext a ⊆ Y , so a ∈ U . We also have U = AU by Proposi-

tion 2.1.3 for and ext.

Suppose that c ∈ S, and Pos(b↓ c) → c ∈ U . For each y ∈ ext c with y ∈ cl(ext b)

we have ext b G ext c by the definition of closure, so Pos(b↓c), hence c ∈ U and y ∈ Y .

Thus ext c ⊆ Y and therefore c ∈ U .

We have shown that U satisfies the conditions of Definition 3.6.1, so S /U . There-

fore Y = X, so cl(ext b) ⊆ ext a.

Complete regularity

Complete regularity is defined for formal topologies in [33]. Like regularity, complete

regularity also depends on the notion of one formal open being well-inside another,

so it seems likely that we can obtain three different notions of complete regularity

corresponding to the three notions of “well-inside”. However we shall leave this as a

possible topic for future research. A definition of weak complete regularity has already

been given by Johnstone in [48] for locales, by replacing “closure” with “weak closure”

in the definition of complete regularity.


3.6.2 Compactness and local compactness

Compactness

As was mentioned in the introduction, it is far harder to prove that topological spaces

are compact constructively than it is classically. For example, the Fan Theorem,

which is equivalent to the assertion that the Cantor space is compact, is inconsistent

with the principles of CRM, and hence is not provable in CZF (this can be seen using

realisability models [58, 70]). In contrast to this, it is often much easier to prove that

certain formal topologies are compact. The standard definition of compactness for

formal topologies is as follows.

Definition 3.6.8 A formal topology S is compact if, given any U ∈ Pow(S) with

AU = S, there is a finite subset U0 ⊆ U such that AU0 = S.

Local compactness

Local compactness for formal topologies has been formulated in several different ways

in the past. The definition that we shall adopt here is taken from [31, 33], and is also

the one used in [7].

Definition 3.6.9 If S is a formal topology and U, V ∈ Pow(S), we say that U is

way below V if

(∀W ∈ Pow(S))[V / W → (∃W0 ∈ FW )[U /W0]]

where FW is the set of finite subsets of W .

We identify elements of S with singleton subsets of S so, for example, if a, b ∈ S

we say that a is way below b if a is way below b.

Definition 3.6.10 A formal topology S is locally compact if for all a ∈ S there is

a subset U ⊆ S such that a / U and b is way below a for all b ∈ U .

It is remarked in [7] that if S is locally compact then there is a function wb : S →

Pow(S) such that a / wb(a) for all a ∈ S, and b is way below a for all a ∈ S and

b ∈ wb(a).

Chapter 4

Inductively Generated Formal

Topologies

4.1 Covering systems and set-presentability

We begin this section by showing how a formal topology can be inductively generated

using a covering system. Covering systems can also be thought of as a way of present-

ing a Frame in terms of a set of generators and a set of relations; we shall make this

notion more precise by proving a universal property of inductively generated formal

topologies.

The ideas of presenting frames using generators and relations and of inductively

generating formal topologies using a collection of “rules” have been around for some

time (see, for example, [45] and [73]). The particular notion of covering system

on which this chapter is based was described in some detail in [27], working in type

theory. As we are working in constructive set theory, we shall instead present covering

systems in the same style as [7], but the two approaches are essentially the same.

We shall also consider the notion of a set-presentable formal topology. It was

shown in [7] that, in the presence of REA, a formal topology is set-presentable if and

only if it is inductively generated.

88

CHAPTER 4. INDUCTIVELY GENERATED FORMAL TOPOLOGIES 89

4.1.1 Covering systems

Definition 4.1.1 A covering system is a triple C = 〈S,≤, C〉 consisting of a set S,

a preorder ≤ on S and a function C : S → Pow(Pow(S)).

A covering system C is local if for all a ∈ S and X ∈ C(a):

• X ⊆↓a

• (∀b ≤ a)(∃Y ∈ C(b))[Y ⊆↓X]

Definition 4.1.2 Given two covering systems C1 = 〈S,≤, C1〉 and C2 = 〈S,≤, C2〉

on the same preordered set S, C1 and C2 are equivalent if for all a ∈ S,

• (∀X1 ∈ C1(a))(∃b ≥ a)(∃X2 ∈ C2(b))[X2 ↓ a ⊆↓X1]

• (∀X2 ∈ C2(a))(∃b ≥ a)(∃X1 ∈ C1(b))[X1 ↓ a ⊆↓X2]

Proposition 4.1.3 Two local covering systems C1 and C2 are equivalent iff, for all

a ∈ S,

• (∀X1 ∈ C1(a))(∃X2 ∈ C2(a))[X2 ⊆↓X1]

• (∀X2 ∈ C2(a))(∃X1 ∈ C1(a))[X1 ⊆↓X2]

Definition 4.1.4 Given a local covering system C = 〈S,≤, C〉, a subclass U ⊆ S is

C-closed if a ∈ U whenever there is an X ∈ C(a) with X ⊆ U .

Given a local covering system C, assuming the Regular Extension Axiom (REA)

we can inductively generate a formal topology IG0(C) on the set S preordered by ≤,

by taking AU to be the smallest class Y such that ↓U ⊆ Y and Y is C-closed. The

following result is proved in [7].

Proposition 4.1.5 (CZF + REA) Given a local covering system C, the induc-

tively defined class AU is a set for all U ∈ Pow(S) and A satisfies axioms A1–A3.

From now on, whenever inductively generated formal topologies are mentioned,

it should be assumed that we are working in CZF + REA.


Lemma 4.1.6 If C1 and C2 are equivalent, then IG0(C1) = IG0(C2).

Proof. Given U ∈ Pow(S), it is easy to prove thatA1U contains ↓U and is C2-closed,

so that A1U ⊆ A2U . Similarly A2U ⊆ A1U .

If C is any covering system (not necessarily local), define IG(C) to be the formal

topology inductively generated by the local covering system C ′ = 〈S,≤, C ′〉, where

C ′(a) = X ↓a | a ≤ b & X ∈ C(b)

for all a ∈ S.

Proposition 4.1.7 (i) C is equivalent to the local covering system C ′ defined above.

(ii) If C1 and C2 are equivalent covering systems, then the local covering systems

C ′1 and C ′2 obtained from C1 and C2 respectively are equivalent and hence IG(C1) =

IG(C2).

(iii) If C is local then IG(C) = IG0(C).

Proof. For (i), observe that if X ∈ C(a) then X ↓ a ∈ C ′(a) and X ↓ a ⊆↓X.

Conversely given Y ∈ C ′(a), Y = X ↓a for some b ≥ a and X ∈ C(b).

It is easy to see that equivalence of covering systems is transitive, so (ii) follows

from (i) and Lemma 4.1.6.

For (iii) we have to show that IG0(C) = IG0(C ′), where C ′ is the local cover-

ing system obtained from C as described above. Again this is follows from (i) and

Lemma 4.1.6.

4.1.2 Morphisms and points of inductively generated formal

topologies

Given a covering system C, we can give an explicit description of the points of IG(C).

Lemma 4.1.8 If C = 〈S,≤, C〉 is a local covering system and α ⊆ S, then α satisfies

the condition FP3 for formal points of IG0(C) (see Definition 3.2.1) if and only if


(i) (∀a ∈ α)(∀X ∈ C(a))[α G X], and

(ii) α = ↑α.

Proof. Suppose that α ⊆ S satisfies FP3, a ∈ α and X ∈ C(a). Then a / X, so

α G X by FP3. If a ∈ α and a ≤ b then a / b, so a G b. Thus α satisfies (i) and

(ii).

Conversely suppose that α satisfies (i) and (ii). Given U ∈ Pow(S), let

Y = a ∈ S | a ∈ α→ α G U

↓U ⊆ Y by (ii), and Y is C-closed by (i), so AU ⊆ Y . Thus α satisfies FP3.

Proposition 4.1.9 If C = 〈S,≤, C〉 is a covering system and α ⊆ S, then α is a

formal point of IG(C) if and only if

FP1: ∃a ∈ α

FP2: (∀a, b ∈ α)(∃c ∈ α)[c ≤ a, b]

FP3a: (∀a ∈ α)(∀X ∈ C(a))[α G X]

FP3b: α = ↑α.

Proof. By Lemma 4.1.8, α satisfies FP3 iff

(i) (∀a ∈ α)(∀b ≥ a)(∀X ∈ C(b))[α G X ↓a], and

(ii) α = ↑α.

FP3b and (ii) are identical. It is easy to see that, in the presence of (ii), (i) implies

FP3a. Conversely if FP2, FP3a and FP3b hold, a ∈ α, a ≤ b and X ∈ C(b), then

b ∈ α, so α G X and hence α G X ↓ a by FP2; thus FP2, FP3a and FP3b imply

(i).

There is a similar result for formal topology maps into an inductively generated

formal topology.


Lemma 4.1.10 If S1 is a formal topology and C2 = 〈S2,≤2, C2〉 is a local covering

system, then a relation r ⊆ S1 × S2 satisfies the axiom FTM3 from Definition 3.1.2

for a formal topology map S1 → IG0(C2) if and only if

(i) (∀a ∈ S2)(∀X ∈ C2(a))[r−a /1 r

−X], and

(ii) (∀a, b ∈ S2)[a ≤2 b→ r−a /1 r−b]

Proof. Suppose that r satisfies FTM3. If a ∈ S2 and X ∈ C2(a) then a /2 X, so

r−a /1 r−X by FTM3. If a ≤2 b then a /2 b and hence r−a /1 r

−b by FTM3.

Now suppose that r satisfies (i) and (ii). Given U ∈ Pow(S2), let

Y = a ∈ S2 | r−a /1 r−U

Then Y is C2-closed by (i), and ↓U ⊆ Y by (ii), so A2U ⊆ Y .

Proposition 4.1.11 If S1 is a formal topology and C2 = 〈S2,≤2, C2〉 is a covering

system, then a relation r ⊆ S1 × S2 is a formal topology map S1 → IG(C2) if and

only if

FTM1: S1 / r−S2

FTM2: (∀a, b ∈ S2)[r−a ∩ r−b ⊆ A1r

−(a↓b)]

FTM3a: (∀a ∈ S2)(∀X ∈ C2(a))[r−a /1 r

−X]

FTM3b: (∀a, b ∈ S2)[a ≤2 b→ r−a /1 r−b]

FTM4: (∀a ∈ S2)[r−a = A1r

−a]

Proof. By Lemma 4.1.10 r satisfies FTM3 if and only if

(i) (∀a ∈ S2)(∀b ≥2 a)(∀X ∈ C2(b))[r−a /1 r

−(X ↓a)], and

(ii) (∀a, b ∈ S2)[a ≤2 b→ r−a /1 r−b].

Note that (ii) is identical to FTM3b. If r satisfies (i) and (ii), a ∈ S2 and X ∈

C2(a), then r−a /1 r−(X ↓a) ⊆ r−(↓X) by (i) and r−(↓X) /1 r

−X by (ii), so r

satisfies FTM3a.


Conversely suppose that r satisfies FTM2, FTM3a, FTM3b and FTM4. Let a, b ∈

S2, a ≤2 b and X ∈ C2(b). Then r−b /1 r−X by FTM3a, and r−a /1 r

−b by

FTM3b. Using FTM4 and FTM2 we have

r−a /1 r−a↓r−b /1 r−a↓r−X = r−a ∩ r−X /1 r−(a↓X)

4.1.3 A universal property of inductively generated formal

topologies

It is shown in [27] that if S = IG(C) then S is the smallest formal topology satisfying

the axioms of C, that is, AU is C-closed for all U ∈ Pow(S), and if S ′ is another

formal topology on the same set S such that A′U is C-closed for each U ∈ Pow(S),

then AU ⊆ A′U for all U ∈ Pow(S). We describe a different universal property

below, which shows that a covering system can also be thought of as a presentation

of a Frame using generators and relations.

Proposition 4.1.12 If C is a covering system on S, A is a Frame and f : S → A

is a function satisfying

(i) a ≤ b→ f(a) ≤ f(b)

(ii) > ≤∨f(a) | a ∈ S

(iii) f(a) ∧ f(b) ≤∨f(c) | c ∈ a↓b

(iv) X ∈ C(a) → f(a) ≤∨f(b) | b ∈ X

then there is a unique Frame homomorphism g : Sat(IG(C)) → A such that f(a) =

g(Aa) for all a ∈ S.

Furthermore the function a 7→ Aa from S to Sat(IG(C)) satisfies conditions

(i)-(iv) above.

Proof. If such a g exists then it must be unique because any Frame homomorphism

on the set-generated Frame Sat(IG(C)) is determined by its values on the generators.

Define a class-function g : Sat(IG(C)) → A by setting g(U) =∨f(a) | a ∈ U for


each U ∈ Sat(IG(C)). We shall show that g is a frame homomorphism and that

f(a) = g(Aa) for all a ∈ S.

Given U ∈ Pow(S), let YU = a ∈ S | f(a) ≤∨f(b) | b ∈ U. ↓ U ⊆ YU

because f preserves ≤. If a ≤ a′ ∈ S, X ∈ C(a′) and X ↓a ⊆ YU , then

f(a) = f(a) ∧ f(a′)

≤ f(a) ∧∨f(b) | b ∈ X since X ∈ C(a′)

=∨f(a) ∧ f(b) | b ∈ X

≤∨f(c) | b ∈ X & c ∈ a↓b using (iii)

=∨f(c) | c ∈ X ↓a

≤∨f(c) | c ∈ YU

≤∨f(b) | b ∈ U since f(c) ≤

∨f(b) | b ∈ U

for all c ∈ YU

and hence a ∈ YU . Thus AU ⊆ YU by induction, so g(AU) =∨f(a) | a ∈ U for all

U ∈ Pow(S).

Clearly U ⊆ V implies g(U) ≤ g(V ) for all U, V ∈ Sat(IG(C)). From the last

paragraph we can see that g(Aa) = f(a) for all a ∈ S. It remains to prove that g

preserves finite meets and all joins.

g preserves finite meets: g(S) = > by (ii), so g preserves the top element. For

binary meets, let U, V ∈ Sat(IG(C)). Then

g(U ↓V ) =∨f(c) | c ∈ U ↓ V

=∨f(c) | a ∈ U & b ∈ V & c ∈ a ↓ b

=∨f(a) ∧ f(b) | a ∈ U & b ∈ V

=∨f(a) | a ∈ U ∧

∨f(b) | b ∈ V

= g(U) ∧ g(V )


g preserves all joins: Let Ui ∈ Sat(IG(C)) for all i ∈ I. Then

g(A

(⋃i∈IUi

))=

∨ f(a) | a ∈

⋃i∈IUi

=

∨i∈I

∨f(a) | a ∈ Ui

=∨

i∈Ig(Ui)

4.1.4 Lindenbaum algebras of geometric theories

Given a set P of propositional symbols, a geometric theory on P is a set of axioms

of the form φ1, . . . , φn ` φ, where φ1, . . . , φn, φ are formulae of the language obtained

by adding > to P and closing under conjunction (∧) and infinitary disjunction (∨

).

Without loss of generality we can assume that for each axiom of T , the premises

φ1, . . . , φn are elements of P and the conclusion φ is of the form∨

i∈I(pi1 ∧ · · · ∧ pi

k(i)),

where each pij ∈ P . Such a theory gives rise to an inductively generated formal

topology, whose points correspond to models of the theory.

Given P , let SP = FP , the set of finite subsets of P , preordered by l1 ≤ l2 iff

l2 ⊆ l1. The elements of SP are to be thought of as finite conjunctions of the propo-

sitional symbols. Given a geometric theory T , define CT : SP → Pow(Pow(SP ))

by

CT (F ) = pi1, . . . , p

ik(i) | i ∈ I | F = p1, . . . , pn &[p1, . . . , pn `

∨i∈I

(pi1 ∧ · · · ∧ pi

k(i))

]∈ T

This defines a covering system CT = 〈SP ,≤, CT 〉. Let ST = IG(CT ), the Lindenbaum

algebra of T .

By Proposition 4.1.9, it is easy to see that the points of ST are precisely those

subsets α ⊆ SP such that

(i) p1, . . . , pn ∈ α iff p1, . . . , pn ∈ α

(ii) If p1, . . . , pn ∈ α and[p1, . . . , pn `

∨i∈I(p

i1 ∧ · · · ∧ pi

k(i))]∈ T then

pi1, . . . , p

ik(i) ∈ α for some i ∈ I.


A model of T is a subset V ⊆ P such that

p1, . . . , pn ∈ V =⇒ (∃i ∈ I)[pi1, . . . , p

ik(i) ∈ V ]

for each axiom p1, . . . , pn `∨

i∈I(pi1 ∧ · · · ∧ pi

k(i)) of T . There is an obvious one-one

correspondence between points of ST and models of T , whereby a point α corresponds

to the model p ∈ P | p ∈ α, and a model V corresponds to the point α = F ∈

SP |F ⊆ V .

Similarly if S is another formal topology, we can describe the formal topology

maps S → ST using Proposition 4.1.11. A relation r ⊆ S × SP is a formal topology

map S → ST iff

(i) r−p1, . . . , pn =S r−p1↓· · ·↓r−pn for all n ≥ 0 and p1, . . . , pn ∈ P

(ii) r−p1, . . . , pn / r−pi1, . . . , p

ik(i) | i ∈ I for each axiom[

p1, . . . , pn `∨

i∈I(pi1 ∧ · · · ∧ pi

k(i))]∈ T

(iii) r−F = Ar−F for all F ∈ SP

There is a one-one correspondence between formal topology maps S → ST and

functions f : P → Sat(S) such that

f(p1)↓· · ·↓f(pn) /⋃i∈I

(f(pi1)↓· · ·↓f(pi

k(i)))

for each axiom[p1, . . . , pn `

∨i∈I(p

i1 ∧ · · · ∧ pi

k(i))]

of T ; these functions can be re-

garded as “interpretations” of T in S.

A formal topology map r : S → ST corresponds to the function f : p 7→ r−p.

Conversely an interpretation f : P → Sat(S) corresponds to the formal topology

map

r = 〈a, p1, . . . , pn〉 ∈ S × SP | a / f(p1)↓· · ·↓f(pn)

Example of a non-spatial formal topology

Lindenbaum algebras can be used to give a constructive example of a formal topology

which is not spatial; this example is based on one by Fourman and Grayson [36].


Given two sets X and Y , the surjective functions X → Y can be axiomatised by

the following geometric theory T on the set P = X × Y :

`∨y∈Y

〈x, y〉 for each x ∈ X

`∨x∈X

〈x, y〉 for each y ∈ Y

〈x, y1〉, 〈x, y2〉 `∨> | y1 = y2 for all x ∈ X and y1, y2 ∈ Y

The models of T are precisely the subsets of X × Y which are graphs of surjections

X → Y , so if X = N and Y = 2N then T has no models, and therefore the formal

topology ST has no formal points. However, it can be verified (for this choice of X

and Y )1 that the set

Pos = f ∈ ST | (∀x ∈ X)(∀y1, y2 ∈ Y )[〈x, y1〉, 〈x, y2〉 ∈ f → y1 = y2]

is a unary positivity predicate for ST . If ST were spatial then Pos would have to be

empty, but we have 〈x, y〉 ∈ Pos for all x ∈ X and y ∈ Y .

4.1.5 Adding a unary positivity predicate

There are two possible approaches if we would like our inductively generated formal

topologies to have a unary positivity predicate. The simplest approach is to induc-

tively generate a formal topology using a covering system as above, and then try to

find a suitable positivity predicate Pos satisfying Monotonicity and Positivity for that

formal topology.

Another approach, taken in [27], is to start with a covering system and a positivity

predicate Pos satisfying certain conditions, and then add extra axioms to the covering

system to ensure that the inductively generated formal topology is open and admits

Pos as its positivity predicate.

Following [27], given a covering system C = 〈S,≤, C〉 and a subset Pos ⊆ S, we

say that Pos is monotone on the axioms of C iff Pos a & X ∈ C(a) → X G Pos for all

1In checking this we found it necessary to use the fact that X is discrete, and that given anyfinite subset F ⊆ Y there is a y ∈ Y such that y /∈ F .


a ∈ S and X ∈ C(a). We say that Pos is monotone on ≤ iff Pos a & a ≤ b → Pos b

for all a, b ∈ S.

Proposition 4.1.13 If C = 〈S,≤, C〉 is a covering system and Pos ⊆ S is monotone

on the axioms of C and on ≤, define C ′ = 〈S,≤, C ′〉 where

C ′(a) = C(a) ∪ a | Pos a

for each a ∈ S. Then Pos is a positivity predicate for IG(C ′).

Note that if C is local then so is C ′.

4.1.6 Set-presentable formal topologies

The definition of a formal topology relies on the use of a proper class, namely the

covering relation /. Set-presentations allow certain formal topologies to be defined

purely in terms of sets. The following definition is taken from [7], where it was also

proved that, in the presence of REA, a formal topology is set-presentable if and only

if it can be inductively generated by a covering system.

Definition 4.1.14 A set-presentation for a formal topology S is a function C : S →

Pow(Pow(S)) such that

a / U ⇔ (∃U0 ∈ C(a))[U0 ⊆ U ]

A formal topology is set-presentable if it has a set-presentation.

In [33] it was proved that if S is set-presentable and S ∼= S ′ then S ′ is set-

presentable. Curi’s proof relied on Type-Theoretic Choice; the reason for the com-

plexity of the proof seems to be that set-presentability of a formal topology is a

property not just of the corresponding Frame, but also of the choice of base.

We will show that in CZF, the use of Choice can be avoided by using the Strong

Collection and Subset Collection schemes.

Lemma 4.1.15 Given a formal topology S, the following are equivalent.


(i) S is set-presentable

(ii) For all a ∈ S there is a set U ∈ Pow(Pow(S)) such that

(∀V ∈ Pow(S))[a / V → (∃V0 ∈ U)[a / V0 & V0 ⊆ V ]]

(iii) For all U ∈ Pow(S) there is a set U ∈ Pow(Pow(S)) such that

(∀V ∈ Pow(S))[U / V → (∃V0 ∈ U)[U / V0 & V0 ⊆ V ]]

Proof. (i) ⇒ (iii): Suppose that S has a set-presentation C : S → Pow(Pow(S)).

Let C =⋃

a∈S C(a). By Subset Collection there is a set D such that for any

set V , if

(∀a ∈ U)(∃W ∈ C)[W ∈ C(a) & W ⊆ V ] (1)

then there is a D ∈ D such that

(∀a ∈ U)(∃W ∈ D)[W ∈ C(a) & W ⊆ V ] (2)

and (∀W ∈ D)(∃a ∈ U)[W ∈ C(a) & W ⊆ V ] (3)

If V ∈ Pow(S) and U / V , then a / V for all a ∈ U and hence (1) holds by the

definition of a set-presentation. So there is a D ∈ D such that (2) and (3) hold,

and hence U /⋃D and

⋃D ⊆ V .

Let U = ⋃D |D ∈ D &

⋃D ⊆ S & U /

⋃D.

(iii) ⇒ (ii) is trivial.

(ii) ⇒ (i): Suppose that (ii) is true. Then

(∀a ∈ S)(∃U)(∀V ∈ Pow(S))[a / V → (∃V0 ∈ U)[a / V0 & V0 ⊆ V ] ]

By Collection there is a set U such that

(∀a ∈ S)(∃U ∈ U)(∀V ∈ Pow(S))[a / V → (∃V0 ∈ U)[a / V0 & V0 ⊆ V ] ]

Define C : S → Pow(Pow(S)) by C(a) = W ∈⋃

U |W ⊆ S & a / W. Then

C is a set-presentation for S.


Lemma 4.1.16 If r : S → S ′ is a formal topology map and U ∈ Pow(S), then there

is a set U ′ such that

(∀U ′ ∈ Pow(S ′))[U ⊆ r−U ′ → (∃U ′0 ∈ U ′)[U ⊆ r−U ′

0 & U ′0 ⊆ U ′]]

Proof. By Subset Collection there is a set U ′ such that for any set U ′, if

(∀a ∈ U)(∃a′ ∈ S ′)[ara′ & a′ ∈ U ′]

then there is a U ′0 ∈ U ′ such that

(∀a ∈ U)(∃a′ ∈ U ′0)[ara

′ & a′ ∈ U ′]

and (∀a′ ∈ U ′0)(∃a ∈ U)[ara′ & a′ ∈ U ′]

The conclusion follows immediately from this.

Proposition 4.1.17 If S and S ′ are formal topologies, S is set-presentable and S ∼=

S ′, then S ′ is set-presentable.

Proof. Let C : S → Pow(Pow(S)) be a set-presentation for S, and let r : S → S ′

and s : S ′ → S give an isomorphism between S and S ′.

Given a′ ∈ S ′, by Lemma 4.1.15 there is a set U ∈ Pow(Pow(S)) such that

(∀V ∈ Pow(S))[r−a′ / V → (∃V0 ∈ U)[r−a′ / V0 & V0 ⊆ V ]]

and by Lemma 4.1.16 (∀U ∈ U)(∃U ′)θ(U,U ′), where θ(U,U ′) is the formula

(∀U ′ ∈ Pow(S ′))[U ⊆ r−U ′ → (∃U ′0 ∈ U ′)[U ⊆ r−U ′

0 & U ′0 ⊆ U ′]]

So by Collection there is a set U′ such that (∀U ∈ U)(∃U ′ ∈ U′)θ(U,U ′). Let V ′ =⋃V ′ ∈

⋃U′ |V ′ ⊆ S.

Suppose that U ′ ∈ Pow(S ′) and a′ /′ U ′. Then r−a′ / r−U ′, so there is a U ∈ U

such that r−a′ / U ⊆ r−U ′. From the previous paragraph there is a U ′ ∈ U′ such

that θ(U,U ′) holds, and so there is a U ′0 ∈ U ′ such that U ⊆ r−U ′

0 and U ′0 ⊆ U ′.

Hence r−a′ / r−U ′0, so a′ =S′ s

−r−a′ /′ s−r−U ′0 =S′ U

′.

Thus for all U ′ ∈ Pow(S ′), if a′ /′ U ′ then there is a U ′0 ∈ V ′ such that a′ /′ U ′

0 and

U ′0 ⊆ U ′, so S ′ is set-presentable by Lemma 4.1.15.


4.2 Constructions on inductively generated formal

topologies

One advantage of set-presentable or inductively generated formal topologies is that

a number of constructions can be performed on them which do not appear to be

possible for arbitrary formal topologies. In this section we shall give a brief survey

of some of these constructions.

It has been known for some time that products, coproducts and equalisers of

inductively generated formal topologies can be constructed [27, 62]. With the ex-

ception of coproducts, none of these constructions have been shown to exist without

assuming that the formal topologies are inductively generated. More recently it was

shown by Palmgren [63] that coequalisers of set-presentable formal topologies can be

constructed in a sufficiently strong form of Martin-Lof Type Theory, and the proof

was adapted to CZF + sRRP-REA by Aczel [6].

Other constructions which are made possible by set-presentability include function

spaces (S1 → S2) where S1 is locally compact and S2 is set-presentable [33], as well

as the upper and lower powerlocales of an inductively generated formal topology

[89, 87, 93]. The upper powerlocale provides a method of proving that a given formal

topology is compact, which we shall apply later to prove that the completion of

a totally bounded uniform formal topology is compact. The lower powerlocale is

of interest because its points correspond to the weakly closed subspaces with open

domain.

Finally we shall show that if S ′ is a subspace of an inductively generated formal

topology S, and S ′ has open domain (i.e. has a unary positivity predicate), then the

weak closure of S ′ in S exists and is an inductively generated formal topology. As

mentioned in the introduction, this duplicates some of the work of Vickers in [93].


4.2.1 Binary products

Binary products of inductively generated formal topologies were shown to exist in

[27].

Proposition 4.2.1 If Ci = 〈Si,≤i, Ci〉 are covering systems for i = 1, 2, then the bi-

nary product of IG(C1) and IG(C2) exists and is inductively generated by the covering

system C = 〈S,≤, C〉, where

• S = S1 × S2

• 〈a1, a2〉 ≤ 〈b1, b2〉 iff a1 ≤1 b1 and a2 ≤2 b2

• C(〈a1, a2〉) = X1 × a2 |X1 ∈ C1(a1) ∪ a1 ×X2 |X2 ∈ C2(a2)

with the projections πi : IG(C) → IG(Ci) given by

πi = 〈〈a1, a2〉, bi〉 ∈ S × Si | ai /i bi

4.2.2 Set-indexed products

Given a set-indexed family of covering systems (Ci)i∈I , let Si = IG(Ci) for each i ∈ I.

The product of (Si)i∈I can be inductively generated as the Lindenbaum algebra of a

geometric theory.

Let P =∐

i∈I Si = 〈i, a〉 | i ∈ I & a ∈ Si. Define a geometric theory T on P ,

whose axioms are the formulae

〈i, a〉 `∨〈i, b〉 | b ∈ X

for each 〈i, a〉 ∈ P and X ∈ Ci(a). Let S be the Lindenbaum algebra of T . For each

i ∈ I, define a formal topology map πi : S → Si by

πi = 〈F, a〉 |F ∈ A〈i, a〉

Proposition 4.2.2 S is the product of (Si)i∈I , and the projections are the formal

topology maps πi : S → Si.


Sketch of proof. Given a formal topology S ′ and formal topology maps ri : S ′ → Si

for all i ∈ I, we have to show that there is a unique map r : S ′ → S such that ri = πir

for each i ∈ I.

From Subsection 4.1.4 we know that there is a one-one correspondence between

formal topology maps r : S ′ → S and functions f : P → Sat(S ′) such that

f(〈i, a〉) /′⋃f(〈i, b〉) | b ∈ X

for all i ∈ I, a ∈ Si and X ∈ Ci(a). It can be shown that for each i ∈ I, ri = πi r if

and only if f(〈i, a〉) = r−i a for all a ∈ S.

For existence, define f : P → Sat(S ′) by f(〈i, a〉) = r−i a. It follows from

FTM3a that f(〈i, a〉) /′⋃f(〈i, b〉) | b ∈ X for all i ∈ I, i ∈ Si and X ∈ Ci(a).

For uniqueness, suppose that s : S ′ → S and ri = πi s for each i ∈ I. Let g :

P → Sat(S ′) be the function corresponding to s. Then f(〈i, a〉) = r−i a = g(〈i, a〉)

for all i ∈ I and a ∈ S, so f = g and hence r = s.

4.2.3 Equalisers

Equalisers are covered in [62]. Suppose that S1 and S2 are two formal topologies,

with S1 generated by a local covering system C1 = 〈S1,≤1, C1〉, and that we are given

a pair of formal topology maps S1

r−→−→sS2.

Let S ′1 be the formal topology inductively generated by the local covering system

C ′1 = 〈S1,≤1, C′1〉, where

C ′1(a) = C1(a) ∪ r−b | b ∈ S2 & asb ∪ s−b | b ∈ S2 & arb

Proposition 4.2.3 S ′1 is the equaliser of S1 and S2, and the equalising morphism is

the subspace inclusion e : S ′1 → S1:

e = 〈a′, a〉 ∈ S1 × S1 | a′ /′1 a

4.2.4 Coequalisers

The impredicative construction of coequalisers of locales, that is, equalisers of frames,

is straightforward: given frame homomorphisms f, g : A → B, their equaliser is the


frame A′ = a ∈ A | f(a) = g(a), with the obvious inclusion e : A′ → A. The

problem is somewhat harder predicatively, where A and B are set-generated Frames,

because it is not easy to show that A′ is set-generated, even in the case where A

arises from an inductively generated formal topology.

The first step towards a solution was taken by Ishihara and Palmgren in [43],

where it was shown how to construct quotients of topological spaces, working in type

theory. To find a base for the topology on the quotient space, a Strong Regularity

Axiom in type-theory was assumed, together with Dependent Choices. It was also

shown how to adapt the proof to the theory CZF + RDC + sREA, and in fact

Palmgren’s proof can be modified to use only CZF + sRRP-REA.

A point-free proof of the existence of coequalisers of set-presentable formal topolo-

gies was given in [63], again working in type theory. As well as using type-theoretic

choice, this proof used type-theoretic universes closed under various operations, in-

cluding the construction of W -types. The proof was later adapted to CZF + sRRP-

REA by Peter Aczel in [6], where it was shown that equalisers of set-generated Frames

can be constructed, provided the corresponding formal topologies are set-presentable;

it follows from this that coequalisers of set-presentable formal topologies exist.

4.2.5 Function spaces

It was shown in [41] that a locale A has function spaces (that is, BA exists for every

locale B) if and only if A is locally compact. More recently in [33], Giovanni Curi

has shown how to construct the function space S1 → S2, where S1 and S2 are formal

topologies with S1 locally compact and S2 set-presentable. We shall describe Curi’s

construction below using Lindenbaum algebras to simplify the presentation.

Proposition 4.2.4 Every locally compact formal topology is set-presentable. Fur-

thermore if C is any set-presentation of S, then a is way-below b if and only if a b,

where a b is the following restricted formula:

a b⇔ (∀X ∈ C(b))(∃X0 ∈ FX)[a / X0]


Proof. The first part is proved in [7], and the second part follows immediately from

the definitions of “way-below” and set-presentations.

Now suppose that S1 is locally compact and that S2 has a set-presentation

D : S2 → Pow(Pow(S2)). Without loss of generality assume for i = 1, 2 that each

base Si has top and bottom elements >i and ⊥i, and binary meet and join operations

∧,∨ : Si × Si → Si. If necessary these can be added to the bases as described in

Subsection 3.3.4.

The function space S1 → S2 can be constructed as the Lindenbaum algebra of a

geometric theory on the set P = S1×S2. The models of the theory will correspond to

formal topology maps from S1 to S2; the intended meaning of an element 〈a, b〉 ∈ P

is that a is way-below r−b.

The axioms of the geometric theory are as follows:

(1) 〈a, b〉 ` 〈a′,>2〉 (a′ >1)

(2) 〈a, b〉 ` 〈a, c〉 (b /2 c)

(3) 〈a, b〉 ` 〈d, b〉 (d /1 a)

(4) 〈a, b〉, 〈a, c〉 ` 〈a′, b ∧ c〉 (a′ a)

(5) 〈a, b〉 `∨

b′∈X∗ 〈a, b′〉 (X ∈ D(b))

(6a) 〈a1, b〉, 〈a2, b〉 ` 〈a1 ∨ a2, b〉

(6b) ` 〈⊥1, b〉

(7) 〈a, b〉 `∨

aa′ 〈a′, b〉

(8a) 〈a, b1 ∨ b2〉 `∨

a/a1∨a2(〈a1, b1〉 ∧ 〈a2, b2〉)

(8b) 〈a,⊥2〉 `∨

a/∅>2

where X∗ is the set of finite joins of elements of X. Axioms (6b) and (8b) were not

present in [33] but appear to be necessary. It can be shown that S1 → S2 satisfies the

properties required of a function space, that is, there is an isomorphism between the

hom-sets FTop(S3×S1,S2) and FTop(S3, (S1 → S2)) natural in S3. In particular it

is shown in [33] that the formal points of (S1 → S2) correspond to formal topology

maps from S1 to S2.


4.2.6 Weakly closed subspaces

We observed in subsection 3.5.4 that given a subspace S ′ v S, it appears that in

general we have no way of constructing the weak closure of S ′ in S predicatively. As

promised, we will show how this can be done in the special case where S is inductively

generated and S ′ is an open formal topology.

By Corollary 3.5.12, if the weak closure of a subspace S ′ with open domain (that

is, which is itself an open formal topology) exists, then it must be determined entirely

by the positivity predicate of S ′.

Proposition 4.2.5 Given a local covering system C = 〈S,≤, C〉 and a subset P ⊆ S

with P = ↑P , define a new covering system CP = 〈S,≤, CP 〉, where

CP (a) = C(a) ∪ a | a ∈ P

for each a ∈ S. Then:

(i) IG(CP ) is a weakly closed subspace of IG(C).

(ii) If a ∈ P → X G P for all a ∈ S and X ∈ C(a), then IG(CP ) is open and P is

its positivity predicate.

Proof. (i) Let S = IG(C), SP = IG(CP ) and suppose that SP v S ′ v S, with the

subspace inclusion SP v S ′ strongly dense. We have to show that APU ⊆ A′U

for all U ∈ Pow(S). For this it suffices to show that A′U contains ↓U and is

CP -closed.

↓U ⊆ AU ⊆ A′U since S ′ is a subspace of S. If a ∈ S, X ∈ C(a) and X ⊆ A′U ,

then a ∈ AX ⊆ A′X ⊆ A′U . Hence A′U is C-closed.

To complete the proof that A′U is CP -closed, suppose that a ∈ S and a | a ∈

P ⊆ A′U , that is, a ∈ P → a /′ U . From the extra axioms added to CP we

have a /P a | a ∈ P, so a /′ a | a ∈ P since S ′ v S is strongly dense. Thus

a ∈ A′U .


(ii) Interestingly this construction is identical to the one used by Coquand, Sambin,

Smith and Valentini in [27] to deal with the addition of a unary positivity

predicate to a covering system (see Proposition 4.1.13).

Note that if S = IG0(C), P ⊆ S, CP is the covering system defined above and

SP = IG0(CP ), then

Pt(SP ) = α ∈ Pt(S) |α ⊆ P = restP

Proposition 4.2.6 If S ′ v S, where S = IG0(S) and S ′ is an open formal topology,

then the weak closure of S ′ in S exists and is given by IG0(CPos′).

Proof. Let SPos′ = IG0(CPos′). By the last Proposition, S ′ and IG0(CPos′) have the

same positivity predicate Pos′, so the inclusion S ′ v SPos′ is strongly dense.

Suppose that S ′ v S ′′ v S and S ′ v S ′′ is strongly dense. Then APos′U ⊆ A′′U

for all U ∈ Pow(S), because S ′′ is a subspace of S and (Pos′ a → a /′ U) → a /′ U .

Thus S ′′ v SPos′ . So SP is the largest subspace of S in which S ′ is strongly dense.

Weak regularity

We are now in a position to justify Definition 3.6.1. Let S be inductively generated by

a local covering system C, and suppose that S is open, with positivity predicate Pos.

For each a ∈ S, let Sa v S be the open subspace of S corresponding to a, described

in Subsection 3.5.2. Sa is an open formal topology, and its positivity predicate is

Posa = b ∈ S | Pos(a↓b)

Let S ′a be the weak closure of Sa in S for each a ∈ S. By Proposition 4.2.6 S ′a is

inductively generated by the covering system C ′a = 〈S,≤, C ′a〉, where

C ′a(b) = C(b) ∪ b | Pos(a↓b)

Proposition 4.2.7 a is 1-well inside b if and only if S ′a v Sb.


Proof. By Proposition 3.5.3, S ′a v Sb if and only if A′ab = S. This holds if and

only if U = S whenever U ∈ Pow(S), b ∈ U , U = ↓U and U is C ′a-closed.

U is C ′a-closed iff U is C-closed and (∀c ∈ S)[[Pos(a↓c) → c ∈ U ] → c ∈ U ]. Thus

S ′a v Sb if and only if U = S whenever b ∈ U , AU = U and (∀c ∈ S)[[Pos(a ↓ c) →

c ∈ U ] → c ∈ U ].

4.2.7 Upper and lower powerlocales

The upper powerlocale

A Preframe is a preordered class P with finite meets and all directed set-joins, sat-

isfying the infinite distributive law

a ∧∨↑

U =∨↑

a ∧ b | b ∈ U

for each a ∈ P and each directed subset U of P (regarding two elements a, b ∈ P

as equal if a ≤ b and b ≤ a). A Preframe homomorphism is an order-preserving

Function between Preframes which preserves the finite meets and directed joins.

The upper powerlocale PUA of a Locale A is defined to be the free Frame on A,

regarded as a Preframe. There is a Preframe map from A into PUA (preserving finite

meets and directed joins), but finite joins are freely added in the construction of PUA.

Vickers [94] has shown how to predicatively construct the upper powerlocale of

an inductively generated formal topology, and in [87] this is used to give a character-

isation of compactness in inductively generated formal topologies.

The base the upper powerlocale of S will be the set FS of finite subsets of S.

Each finite subset a1, . . . , an is to be interpreted as the join of a1, . . . , an in the

original frame Sat(S). The preorder vL on FS is defined by F vL G iff (∀a ∈

F )(∃b ∈ G)[a ≤ b].

Theorem 4.2.8 (Vickers) The upper powerlocale of S = IG0(〈S,≤, C〉) is given

by PUS = IG0(〈FS,vL, CU〉), where CU : FS → Pow(Pow(FS)) is defined by

CU(F ) = F(X1 ∪ · · · ∪Xn∪ ↓G) |G ∈ FF & F = a1, . . . , an ∪G

& X1 ∈ C(a1), . . . , Xn ∈ C(an)


Proof. See [94].

The points of the upper powerlocale are considered, in the impredicative case, by

Vickers in [89]. The main result which will be used in this thesis is the following:

Theorem 4.2.9 (Vickers) If S = IG0(〈S,≤, C〉) is an inductively generated formal

topology then the following are equivalent:

(i) S is compact.

(ii) There is a point θ of PUS such that (∀F ∈ θ)[S / F ]

(iii) There is a subset θ ⊆ FS such that

1. if F ∈ θ and F vL G then G ∈ θ

2. if F ∈ FS, a∪F ∈ θ and X ∈ C(a) then G∪F ∈ θ for some G ∈ FX.

3. ∃F ∈ θ

4. if F ∈ θ then S / F

If (ii) or (iii) holds for θ, then θ = F ∈ FS |S / F.

Proof. See [87] for a direct proof of the equivalence between (i) and (iii), and the

fact that θ = F ∈ FS |S / F.

It is easy to see from Proposition 4.1.9 that if θ is a formal point of PU satisfying

(ii) then it must satisfy the conditions of (iii). Conversely if θ satisfies the equivalent

conditions (i) and (iii) then it satisfies (ii) (in particular FP2 follows from A3 and

(i)).

The lower powerlocale

Impredicatively, the lower powerlocale of a locale A is the free frame on A, regarded

as a∨

-semilattice. Working in CZF, given an inductively generated formal topology

S = IG0(C), where C = 〈S,≤, C〉 is a local covering system, we can obtain the lower


powerlocale of S by using the Lindenbaum algebra construction of Subsection 4.1.4

to freely add finite meets to S.

Define a geometric theory as follows. Let P = S, and let T be the set of all axioms

of the forms

a a b where a, b ∈ S and a ≤ b

and a a∨X where a ∈ S and X ∈ C(a)

Let PLS be the Lindenbaum algebra of T . The base of PLS is the set FS of

finite subsets of S. Each set F ∈ FS can be thought of as the meet of its elements

in the lower powerlocale, in contrast to the upper powerlocale construction where it

corresponded to a finite join in the original formal topology.

Given another formal topology S ′, there is a one-one correspondence between

formal topology maps S ′ → PLS and functions f : S → Sat(S ′) satisfying the

axioms of T . It can be shown that these functions correspond in a natural way to

the∨

-Semilattice homomorphisms Sat(S) → Sat(S ′).

We also noted in Subsection 4.1.4 that the points of PLS correspond to “models”

of T , that is, subsets α ⊆ S such that

(i) α = ↑α, and

(ii) (∀a ∈ α)(∀X ∈ C(a))[α G X]

By Propositions 4.2.5 and 4.2.6, there is a bijection between these “models” and the

weakly closed subspaces of S with open domain (that is, those which are open formal

topologies in their own right). The model α corresponds to the subspace IG(Cα)

defined in Proposition 4.2.5, and if S ′ v S is a weakly closed subspace with open

domain then its positivity predicate is a model of T .

The correspondence between points of the lower powerlocale and weakly closed

subspaces with open domain was first described in the impredicative intuitionistic case

by Bunge and Funk [18] (see also Vickers [89]). The correspondence in predicative

formal topology between the “lower powerpoints” of an inductively generated formal

topology and its weakly closed subspaces with open domain was also observed by

Vickers in [93].


4.3 Application to a result of Coquand

In several of Coquand’s papers transition systems are used to obtain point-free ver-

sions of classical results from combinatorics. These can be viewed as a special class

of covering systems, and have the property that they are spatial in the presence of

classical logic and the axiom of choice.

The use of transition systems in this way can be seen in [22] and [24]. In [22],

a notion of one subset of a transition system being a bar on another subset is used.

This generalises in a natural way to arbitrary covering systems, but the topological

meaning of this notion is not immediately obvious. Based on the earlier work in this

chapter, we have been able to show that U is a bar on V if and only if U covers a

certain weakly closed subspace with open domain, which is determined by V .

In [22], a point-free version of the Open Induction Principle from [69] is given.

We will show how the same result can be expressed using covers of weakly closed

subspaces in place of bars on subsets.

4.3.1 Transition systems and bars

A transition system is an ordered pair 〈S,−−-〉, where S is a set and −−- is a binary

relation on S. Given a, b ∈ S, call b a successor of a if a−−-b. The following definitions

are taken from [22].

Definition 4.3.1 Given a transition system 〈S,−−-〉, a subset U ⊆ S is

• monotone if b ∈ U whenever a ∈ U and a−−-b,

• hereditary if a ∈ U whenever each successor of a is in U .

The hereditary closure of a subset U ⊆ S is the smallest hereditary subclass of

S which contains U . This can be obtained by a simple inductive definition, and

assuming REA the hereditary closure of a subset of S is always a set. A monotone

subset of S is a bar on S if its hereditary closure is the whole of S.

More generally, given U, V ∈ Pow(S), U is hereditary on V if a ∈ U whenever

a ∈ V and every successor of a which is in V lies in U . The hereditary closure of


U in V is the smallest subclass of S which contains U and is hereditary on V . A

monotone subset U of S is a bar on V if the hereditary closure of U ∩ V in V is the

whole of V .

Covering systems arising from transition systems

Given a transition system 〈S,−−-〉, we can obtain a local covering system C = 〈S,≤, C〉,

where ≤ is the reflexive and transitive closure of the converse of −−- (that is, a ≤ b

iff there is a finite sequence b = a0−−-a1−−- · · · −−-an = a, with n ≥ 0), and C : S →

Pow(Pow(S)) is defined by

C(a) = b ∈ S | a−−-b

for each a ∈ S. Let S be the formal topology inductively generated by C.

Proposition 4.3.2 A subset α ⊆ S is a formal point of S if and only if

• α is inhabited

• if a, b ∈ α then there is a c ∈ α together with two finite sequences

a = a0−−- · · · −−-am = c and b = b0−−- · · · −−-bn = c

• if a ∈ α then a−−-b for some b ∈ α

• if b ∈ α and a−−-b then a ∈ α

Proposition 4.3.3 (ZF + DC) The formal topology arising from a transition sys-

tem in this way is spatial.

Proof. Suppose that a ∈ S, U ∈ Pow(S) and every formal point containing a

contains some element of U . Suppose also that a /∈ AU . Let a0 = a. Given ai ∈ S

with ai /∈ AU , since AU is C-closed there must be an ai+1 ∈ S such that ai−−-ai+1

and ai+1 /∈ AU . Using Dependent Choice we can obtain an infinite sequence

a = a0−−-a1−−-a2−−- · · ·


with ai /∈ AU for each i ∈ N. Let α = b ∈ S | (∃i ∈ N)[ai ≤ b]. It is easy to verify

that α is a formal point of S. a ∈ α, so b ∈ α for some b ∈ U . But ai ≤ b for some

i ∈ N, so ai ∈↓U , contradicting ai /∈ AU .

It is easy to see that a subset U ⊆ S is monotone if and only if U = ↓ U and

hereditary if and only if it is C-closed. The hereditary closure of ↓U is the set AU ,

and ↓U is a bar if and only if AU = S. This gives a clear topological interpretation

of the notion of bars in terms of open covers; however, the notion “U is a bar on V ”

is slightly harder to understand in topological terms.

Given a local covering system C = 〈S,≤, C〉 and a subset V ⊆ S, we can obtain

another local covering system C ′V = 〈S ′V ,≤, C ′V 〉, where S ′V = V and C ′

V : V →

Pow(Pow(V )) is defined by

C ′V (a) = X ∩ V |X ∈ C(a)

for each a ∈ V . Given a subset U ⊆ S, we shall say that U is a bar on V if

A′V (U ∩ V ) = V . In the case where C arises from a transition system and U is

monotone, this coincides with the transition system definition of “U is a bar on V ”.

Bars and weakly closed subsets

In [22], the notion of U being a bar on V is only used when V = ↑V . In this case, we

can show that the formal topology S ′V is isomorphic to the weakly closed subspace

SV v S, defined in Subsection 4.2.6. Recall that SV is inductively generated by the

local covering system CV = 〈S,≤, CV 〉, where CV (a) = C(a) ∪ a ∩ V for each

a ∈ S. Note that, since a /V a ∩ V for all a ∈ S, we have U /V U ∩ V for all

U ∈ Pow(S).

Define two relations r ⊆ SV × S ′V and s ⊆ S ′V × SV by

r = 〈a, a′〉 ∈ S × V | a /V a′

and s = 〈a′, a〉 ∈ V × S | a′ /′V (a ∩ V )

r−U ′ =SVU ′ for each U ′ ∈ Pow(V ), and s−U =S′V U ∩ V for each U ∈ Pow(S).


Proposition 4.3.4 r and s are formal topology maps SV → S ′V and S ′V → SV

respectively, and form an isomorphism between SV and S ′V .

Proof. Clearly r and s both satisfy FTM4; the remaining conditions of Proposition

3.1.5 are proved below.

r satisfies FTM3: If a′ ≤ b′ in V then a′ ≤ b′ in S and so AV a′ ⊆ AV b′. Thus

r−a′ ⊆ r−b′.

Now suppose that a′ ∈ V and X ′ ∈ C ′V (a′), so X ′ = X ∩V for some X ∈ C(a′).

Then a′ /V X /V X ∩ V , so

r−a′ /V a′ /V X ∩ V = X ′ /V r−X ′

s satisfies FTM3: If a ≤ b in S then a ∩ V ⊆ b ∩ V , since V = ↑ V . Hence

A′V (a ∩ V ) ⊆ A′

V (b ∩ V ), so s−a ⊆ s−b.

Now let a ∈ S and X ∈ CV (a). We have to show that s−a /′V s−X. Either

(i) X ∈ C(a) or (ii) X = a ∩ V .

(i) s−a =S′V a ∩ V and s−X =S′V X ∩ V . If a ∈ V then X ∩ V ∈ C ′V (a), so

a /′V X ∩ V . Thus a ∩ V /′V X ∩ V .

(ii) s−a =S′V a ∩ V =S′V s−(a ∩ V ).

r−s−a =SVa: r−s−a =SV

a ∩ V =SVa.

s−r−a′ =S′V a′: If a′ ∈ V then s−r−a′ =S′V a

′ ∩ V = a′.

Corollary 4.3.5 Given U, V ∈ Pow(S) with V = ↑V , U is a bar on V if and only

if AVU = S.

Proof. U is a bar on V if and only if S ′V /′V s

−U , which in turn is true if and only if

r−S ′V /V r−s−U . But r−S ′V =SV

S and r−s−U =SVU .

The points of the subspace SV v S are precisely those of the closed subset restV ⊆

Pt(S). If U is a bar on V then restV ⊆ extU in Pt(S). If classical logic is assumed


and S is spatial (for example in ZFC if S arises from a transition system) then SV is

spatial, so U is a bar on V if and only if restV ⊆ extU in Pt(S).

4.3.2 The Open Induction Principle

The point-sensitive Open Induction Principle

Given a set P , set X be the set of infinite sequences on P , written x = (pn)n∈N.

X can be given a topology by taking S to be the set of finite words on P (written

a = p1p2 . . . pn, with n ≥ 0, and denoting the empty word by ∗ ∈ S), and setting

x a if and only if the word a is a prefix of the sequence x. X = 〈X, , S〉 is a small

ct-space.

Suppose also that we are given a binary relation ≺ on P which is well-founded,

that is, if Q ⊆ P and (∀q ≺ p)[q ∈ Q] → p ∈ Q for all p ∈ P , then Q = P . ≺ can be

extended to a binary relation on X by setting (pn)n∈N ≺ (qn)n∈N if and only if there

is an n ∈ N such that pn ≺ qn and pi = qi for all i < n.

It is observed in [69] and [22] that ≺ is not well-founded on X. However, induction

can be used to prove properties corresponding to open subsets of X, if classical logic

and the axiom of choice are assumed. The point-sensitive Open Induction Principle

is as follows, in the form given in [22].

Theorem 4.3.6 (ZFC) If Y ⊆ X is open and (∀y ≺ x)[y ∈ Y ] → x ∈ Y for all

x ∈ X, then Y = X.

The point-free result

The ct-space X can be obtained as the space of formal points of a formal topology

arising from a transition system on S. Given a, b ∈ S, let

a−−-b⇔ (∃p ∈ P )[b = ap]

and let S be the formal topology arising from 〈S,−−-〉, inductively generated by the

covering system C = 〈S,≤, C〉 described in Subsection 4.3.1.


Proposition 4.3.7 The map x 7→ αx is a homeomorphism from X to Pt(S).

The relation ≺ can be extended to a binary relation on S by setting p1 . . . pm ≺

q1 . . . qn if and only if 1 ≤ m ≤ n, pi = qi for all i < m and pm ≺ qm. Given x, y ∈ X,

we have

x ≺ y ⇔ (∃a ∈ αx)(∃b ∈ αy)[a ≺ b]

If Y is an open subset of X then Y = extU for some U ∈ Pow(S). The Open

Induction Principle can be rephrased as

Z(U) ⊆ extU ⇒ X ⊆ extU

where Z(U) = x ∈ X | (∀y ∈ X)[y ≺ x → y ∈ extU ]. Z(U) can be expressed in

terms of the relation ≺ on S:

Z(U) = x ∈ X | (∀y ∈ X)[(∃a ∈ αx)(∃b ∈ αy)[b ≺ a] → y ∈ extU ]

= x ∈ X | (∀y ∈ X)(∀a ∈ αx)(∀b ∈ αy)[b ≺ a→ y ∈ extU ]

= x ∈ X | (∀a ∈ αx)(∀b ∈ S)(∀y ∈ ext b)[b ≺ a→ y ∈ extU ]

= restV (U)

where

V (U) = a ∈ S | (∀b ∈ S)(∀y ∈ ext b)[b ≺ a→ y ∈ extU ]

= a ∈ S | (∀b ∈ S)[b ≺ a→ ext b ⊆ extU ]

A point-free version of the Open Induction Principle can now be obtained by

replacing the inclusion ext b ⊆ extU in the definition of V (U), and the two inclusions

restV (U) ⊆ extU and X ⊆ extU , with suitable point-free notions in the formal

topology S. In [22], restV (U) ⊆ extU was replaced by “U is a bar on V (U)”.

Instead we shall use the equivalent statement based on weakly closed subspaces,

based on Corollary 4.3.5. The point-free result is as follows:

Theorem 4.3.8 For each subset U ∈ Pow(S), let

V ′(U) = a ∈ S | (∀b ∈ S)[b ≺ a→ b / U ]


and let SV ′(U) be the weakly closed subspace of S determined by V ′(U)2. If AV (U)U = S

then AU = S.

This can be proved using Coquand’s proof from [22] together with Corollary 4.3.5.

We shall give a more direct proof below.

Lemma 4.3.9 The relation ≺ is well-founded on S: if U ⊆ S and (∀b ≺ a)[b ∈

U ] → a ∈ U for each a ∈ S, then U = S.

Proof. Suppose that (∀b ≺ a)[b ∈ U ] → a ∈ U for all a ∈ S. Given a ∈ S, we shall

prove that a ∈ U by induction on the length of a.

If a has length 0 then there is no b ∈ S for which b ≺ a, so (∀b ≺ a)[b ∈ U ] and

hence a ∈ U .

Now suppose that every word of length ≤ n is in U . Each word of length n+ 1 is

of the form ap, where a has length n and p ∈ P . Let a be a word of length n, and let

Q = p ∈ P | ap ∈ U

Suppose that p ∈ P , and that q ∈ Q for all q ≺ p. If b ≺ ap then either b ≺ a, in

which case b has length ≤ n and is therefore in U , or b = aq for some q ≺ p, in which

case q ∈ Q so b ∈ U . Thus (∀b ≺ ap)[b ∈ U ], so ap ∈ U and hence p ∈ Q.

We have shown that (∀q ≺ p)[q ∈ Q] → p ∈ Q for each p ∈ P , so Q = P by

well-founded induction, and hence ap ∈ U for all p ∈ P .

Lemma 4.3.10 Given U ∈ Pow(S), let

W (U) = a ∈ S | a ∈ V ′(U) → a / U

Then AV ′(U)U ⊆ W (U).

Proof. Clearly ↓U ⊆ W (U). We have to show that W (U) is CV ′(U)-closed. Suppose

that a ∈ S, X ∈ CV ′(U)(a) and X ⊆ W (U). Either (i) X = ap | p ∈ P or

(ii) X = a ∩ V ′(U).

2Note that V ′(U) = ↑V ′(U) since a ≺ b and c ≤ b imply a ≺ c


(i) Suppose that ap ∈ W (U) for all p ∈ P and let a ∈ V ′(U). We have to show that

a/U ; for this it suffices to show that ap/U for all p ∈ P , since AU is C-closed.

This can be done by well-founded induction on P : suppose that p ∈ P and

aq / U for all q ≺ p. We must show that ap / U .

If b ≺ ap then either b ≺ a or b = aq for some q ≺ p. If b ≺ a then b / U since

a ∈ V ′(U), and if b = aq and q ≺ p then b / U by the induction hypothesis.

Thus ap ∈ V ′(U), so ap / U since ap ∈ W (U).

(ii) Suppose that a ∩ V ′(U) ⊆ W (U). a ∈ V ′(U) implies a ∈ W (U) and hence

a / U . Thus a ∈ W (U).

Proof of Theorem 4.3.8. Suppose that AV ′(U)U = S. By Lemma 4.3.10 we have

W (U) = S, that is, (∀b ≺ a)[b / U ] implies a / U for all a ∈ S. Hence AU = S by

Lemma 4.3.9.

4.4 Generalisation to quasi-formal topologies

There are two situations in which we may wish to generate a quasi-formal topology

rather than an “ordinary” formal topology. Firstly, if we are not using the Regular

Extension Axiom then we cannot show thatAU is a set for all U ∈ Pow(S). Secondly,

we may wish to consider covering systems in which the collection of covers does not

form a set.

Definition 4.4.1 A class-covering system consists of a set S with a Preorder ≤,

together with a set-indexed family (Ca)a∈S of subclasses of Pow(S).

A class-covering system is local if for all a ∈ S and X ∈ Ca,

• X ⊆↓a

• (∀b ≤ a)(∃Y ∈ Cb)[Y ⊆↓X]


Inductively generated quasi-formal topologies

Given a local class-covering system (Ca)a∈S, a subclass Y ⊆ S is C-closed if a ∈ Y

whenever a ∈ S, X ∈ Ca and X ⊆ Y . For each U ∈ Pow(S), let AU be the smallest

C-closed class Y such that ↓U ⊆ Y , and let a / U iff a ∈ AU . We will show that S,

≤ and / form a quasi-formal topology.

It is easy to prove that AU = ↓(AU) and that A satisfies the axioms A1 and A2.

Before proving A3 we will need a lemma.

Lemma 4.4.2 For any subclass U ⊆ S,

A+U = a ∈ S | (∃U0 ∈ Pow(U))[a / U ]

is the smallest C-closed class containing ↓U .

Proof. If a ≤ b ∈ U then a ∈ Ab ⊆ A+U . So ↓U ⊆ A+U .

If a ∈ S, X ∈ Ca and X ⊆ A+U , then

(∀b ∈ X)(∃V )[b / V & V ⊆ U ]

so by Strong Collection there is a set V such that

(∀b ∈ X)(∃V ∈ V)[b ∈ AV & V ⊆ U ]

and (∀V ∈ V)(∃b ∈ X)[b ∈ AV & V ⊆ U ]

so taking V =⋃V we have X ⊆ AV and V ⊆ U , so a ∈ AV ⊆ A+U . Thus A+U is

C-closed.

Now suppose that Y is a C-closed class containing ↓U . If a ∈ A+U then a ∈ AV

for some V ∈ Pow(U). Y is a C-closed class containing ↓V , so AV ⊆ Y and hence

a ∈ Y . Thus A+U ⊆ Y .

The following double-induction principle will also be needed:

Lemma 4.4.3 Given a class-covering system (Ca)a∈S, two subsets U, V ∈ Pow(S)

and a subclass Z ⊆ S × S, if


(i) ↓U×↓V ⊆ Z

(ii) (∀a, b ∈ S)(∀X ∈ Cb)[a ×X ⊆ Z → 〈a, b〉 ∈ Z]

(iii) (∀a, b ∈ S)(∀X ∈ Ca)[X × b ⊆ Z → 〈a, b〉 ∈ Z]

then AU ×AV ⊆ Z.

Proof. For each a ∈ S, let Ya = b ∈ S | 〈a, b〉 ∈ Z. Let

Y = a ∈ S | a × AV ⊆ Z = a ∈ S | AV ⊆ Ya

We have to show that AU ⊆ Y ; this can be done by proving that Y contains ↓U and

is C-closed.

To show that ↓U ⊆ Y , suppose that a ≤ a′ ∈ U . It follows from (i) that ↓V ⊆ Ya,

and from (ii) that Ya is C-closed, so AV ⊆ Ya and hence a ∈ Y .

To show that Y is C-closed, let a ∈ S, X ∈ Ca and X ⊆ Y , that is, X×AV ⊆ Z.

If b ∈ AV then X × b ⊆ Z and so 〈a, b〉 ∈ Z by (iii). Thus a × AV ⊆ Z, so

a ∈ Y .

Proposition 4.4.4 A satisfies A3, so S, ≤ and / form a quasi-formal topology.

Proof. Given U, V ∈ Pow(S), let Z = 〈a, b〉 ∈ S × S | a↓ b ⊆ A+(U ↓V ). We will

show that Z satisfies conditions (i)–(iii) of Lemma 4.4.3.

(i) If 〈a, b〉 ∈ ↓U×↓V then a↓b ⊆ U ↓V ⊆ A+(U ↓V ).

(ii) Suppose that a, b ∈ S, X ∈ Cb and a × X ⊆ Z. For each c ∈ a ↓ b, there is

a Y ∈ Cc such that Y ⊆↓X. Y ⊆ a ↓X ⊆ A+(U ↓V ), so c ∈ A+(U ↓V ) by

Lemma 4.4.2.

(iii) By symmetry.

AU × AV ⊆ Z by Lemma 4.4.3, so if a ∈ AU ∩ AV then 〈a, a〉 ∈ Z and hence

a ∈ A+(U ↓V ).


Every quasi-formal topology can be inductively generated

In contrast to ordinary formal topologies and covering systems, every quasi-formal

topology can be inductively generated by a local class-covering system. It is hoped

that this will allow some of the constructions of the previous section to be carried

out on arbitrary quasi-formal topologies, without the need for REA. One possible

application of this is that given a (quasi-)formal topology S, we may be able to give

a meaningful definition of a formal topology map S × S → S, which would be a

necessary first step towards developing a predicative theory of localic groups.

Proposition 4.4.5 Given a quasi-formal topology S, let

Ca = X ∈ Pow(S) |X ⊆↓a & a / X

for each a ∈ S. Then (Ca)a∈S is a local class-covering system which inductively

generates S.

Proof. To show that (Ca)a∈S is local, let a ∈ S and X ∈ Ca. X ⊆↓ a by the

definition of Ca. If b ≤ a then b ∈ A+(b↓X) by A3, so there is a subset Y ⊆ b↓X

such that b ∈ AY . Y ∈ Cb and Y ⊆↓X.

Given U ∈ Pow(S), by A1 we have ↓U ⊆ AU and it follows from A2 that AU is

C-closed.

Now suppose that Y is a C-closed class containing ↓ U . If a ∈ AU then a ∈

A+(a↓U) by A3, so a/X for some subset X ⊆ a↓U . X ∈ Ca and X ⊆↓U ⊆ Y ,

so a ∈ Y . Thus AU ⊆ Y .

4.5 Coinductively generating a binary positivity

predicate

The problem of generalising the method of inductively generating formal topologies

to allow for the generation of a binary positivity predicate has been considered by

Sambin in [72], and by Valentini in [84] and [85]. The addition of the binary positivity


predicate requires a different technique, known as coinduction. WhereasAU is defined

to be the smallest class containing U satisfying certain closure conditions, JU will

be defined as the largest subclass of U which satisfies some other rules.

One difference between the approaches in [72] and in [84, 85] is that in [72] the

binary positivity predicate n is determined only by the covering system for /, while

in [84, 85] the addition of extra axioms for n is allowed.

4.5.1 Coinductive definitions

Given a class S and a subclass Φ ⊆ Pow(S)×S, a subclass B ⊆ S is Φ-progressive if

a ∈ B → X G B for each 〈X, a〉 ∈ Φ. Coinduction is the problem of finding the largest

Φ-progressive subclass of S, or, more generally, the largest Φ-progressive subclass of

some class U ⊆ S. This problem is dealt with in [4], and requires the principles RRS

and RRP-REA in addition to the usual axioms of CZF (see Appendix A). The main

results from [4] are stated and proved below.

Let Φa = X ∈ Pow(S) | 〈X, a〉 ∈ Φ for each a ∈ S, and let ΦU =⋃

a∈U Φa for

each U ∈ Pow(S).

Lemma 4.5.1 (CZF + RRS) If Φa is a set for all a ∈ S, B is a Φ-progressive

subclass of S and b ∈ B, then there is a Φ-progressive set U ∈ Pow(B) such that

b ∈ U .

Proof. Given a set U ∈ Pow(B), sinceB is Φ-progressive we have (∀X ∈ ΦU)(∃c)[c ∈

X ∩B]. By Strong Collection there is a set V such that

(∀X ∈ ΦU)(∃c ∈ V )[c ∈ X ∩B]

and (∀c ∈ V )(∃X ∈ ΦU)[c ∈ X ∩B]

so V ∈ Pow(B) and (∀X ∈ ΦU)[X G V ]. Thus

(∀U ∈ Pow(B))(∃V ∈ Pow(B))(∀X ∈ ΦU)[X G V ]

Now let

R = 〈U, V 〉 ∈ Pow(B)× Pow(B) | (∀X ∈ ΦU)[X G V ]


b ∈ Pow(B) and (∀U ∈ Pow(B))(∃V ∈ Pow(B))[〈U, V 〉 ∈ R], so by RRS there is

a set U ∈ Pow(Pow(B)) such that b ∈ U and (∀U ∈ U)(∃V ∈ U)[〈U, V 〉 ∈ R]. It

follows that⋃U is Φ-progressive, b ∈

⋃U and

⋃U ∈ Pow(B).

Proposition 4.5.2 (CZF + RRS) Given a class S, a subclass Φ ⊆ Pow(S) × S

and a subclass T ⊆ S, the class

J = a ∈ T | (∃U ∈ Pow(T ))[a ∈ U & U is Φ-progressive]

is the largest Φ-progressive subclass of T .

Proof. If a ∈ J and X ∈ Φa, then a ∈ U for some Φ-progressive U ∈ Pow(T ), so

X G U and hence X G J . Thus J is Φ-progressive.

Now suppose that B is a Φ-progressive subclass of T , and let a ∈ B. By Lemma

4.5.1 there is a Φ-progressive set U ∈ Pow(B) ⊆ Pow(T ) such that a ∈ U , so

a ∈ J .

Proposition 4.5.3 (CZF + sRRP-REA) If S and Φ are sets and T ∈ Pow(S)

then the class J defined in Proposition 4.5.2 is a set.

Proof. Let A be a strongly RRP-regular set containing a and Φa for each a ∈ S.

Since A is regular, it follows that ΦU ∈ A whenever U ∈ A and U ⊆ S. We claim

that

J = a ∈ T | (∃U ∈ A)[a ∈ U ⊆ T & U is Φ-progressive]

which is a set by Restricted Separation. The inclusion RHS ⊆ LHS is trivial.

For LHS ⊆ RHS, suppose that a ∈ J , that is, a ∈ B for some Φ-progressive

subset B of T . Let

A′ = U ∈ A |U ⊆ B

If U ∈ A′ then (∀X ∈ ΦU)(∃c)[c ∈ X G B] since U ⊆ B and B is Φ-progressive. Let

r = 〈X, c〉 ∈ ΦU × A | c ∈ X ∩B


r : ΦU > A, so by the regularity of A there is a V ∈ A such that r : ΦU > < V ,

and hence V ⊆ B and (∀X ∈ ΦU)[X G V ]. Thus

(∀U ∈ A′)(∃V ∈ A′)(∀X ∈ ΦU)[X G V ]

Now let

R = 〈U, V 〉 ∈ A′ × A′ | (∀X ∈ ΦU)[X G V ]

We have a ∈ A′ and (∀U ∈ A′)(∃V ∈ A′)[〈U, V 〉 ∈ R]. Since A has the Relation

Reflection Property there is a U ∈ A such that a ∈ U ⊆ A′ and (∀U ∈ U)(∃V ∈

U)[〈U, V 〉 ∈ R]. It follows that a ∈⋃U ⊆ B and

⋃U is Φ-progressive. Also

⋃U ∈ A

since A is union-closed, so a ∈ RHS.

4.5.2 Balanced covering systems

Definition 4.5.4 A balanced covering system is a 4-tuple C = 〈S,≤, C,D〉, where

≤ is a preorder on S and C and D are functions S → Pow(Pow(S)). The balanced

covering system is local if 〈S,≤, C〉 is a local covering system.

We shall only consider local balanced covering systems in this thesis. The function

C : S → Pow(Pow(S)) can be used to inductively generate a covering relation / on

S. The coinductively generated binary positivity predicate n must be compatible

with /, so the rules of C will be used in defining n. D can be used to specify extra

axioms which must be satisfied by n; if D(a) = ∅ for all a ∈ S then n will be the

largest binary positivity predicate which is compatible with /.

Given a subclass U ⊆ S, we shall say that U is C,D-compatible if

(∀a ∈ U)(∀X ∈ C(a) ∪D(a))[X G U ]

Proposition 4.5.5 (CZF + RRS + sRRP-REA) Given a local balanced cover-

ing system C = 〈S,≤, C,D〉, for each U ∈ Pow(S) let AU be the smallest C-closed

class containing ↓U and let JU be the largest subclass of U which is a C,D-compatible

upper section.

AU and JU are sets for each U ∈ Pow(S), and they satisfy the axioms A1–A3,

J1–J2 and C of Definition 3.2.13.


Proof. We already know that AU exists and is a set for each U ∈ Pow(S), and that

A satisfies axioms A1–A3. JU is the largest Φ-progressive subclass of ↑∗U , where

Φ = 〈X, a〉 | a ∈ S & X ∈ C(a) ∪D(a)

∪ 〈b, a〉 | a, b ∈ S & a ≤ b

JU exists by Proposition 4.5.2, and is a set by Proposition 4.5.3. Note that the first

part of the definition of Φ ensures that JU is C,D-compatible, and the second part

ensures that JU = ↑JU .

J1 is true by the definition of JU . For J2, if JU ⊆ V then JU is a Φ-progressive

subclass of V , and J V is the largest Φ-progressive subclass of V , so JU ⊆ J V .

For Compatibility, given U, V ∈ Pow(S) let Y = a ∈ S | a ∈ J V → U G J V .

Since J V = ↑ J V , it follows that ↓ U ⊆ Y . Suppose that a ∈ S, X ∈ C(a) and

X ⊆ Y . If a ∈ J V were true, then we would have X G J V , so that Y G J V and

therefore U G J V . Thus a ∈ Y . We have shown that Y is a C-closed set containing

↓U , so AU ⊆ Y and hence AU G J V → U G J V .

4.5.3 Points and morphisms

Proposition 4.5.6 Let S be a balanced formal topology (co)inductively generated by

C = 〈S,≤, C,D〉. A subset α ⊆ S is a formal point of S if and only if

FP1: ∃a ∈ α

FP2: (∀a, b ∈ α)(∃c ∈ α)[c ≤ a, b]

FP3′a: (∀a ∈ α)(∀X ∈ C(a) ∪D(a))[α G X]

FP3b: α = ↑α

Proof. We must show that α = Jα if and only if α satisfies FP3′a and FP3b above.

But α = Jα if and only if α is Φ-progressive, and this can readily be seen to be

equivalent if and only if FP3′a and FP3b are satisfied.


Comparing this to Proposition 4.1.9, it is clear that if D(a) = ∅ for all a ∈ S

then the formal points of S are precisely those of S regarded as a formal topology

without n.

Proposition 4.5.7 Suppose that S is a balanced formal topology and that C ′ =

〈S ′,≤′, C ′, D′〉 is a local balanced covering system which (co)inductively generates

S ′. A relation r ⊆ S × S ′ is a balanced formal topology map S → S ′ if and only if

FTM1: S / r−S ′

FTM2: (∀a′, b′ ∈ S ′)[r−a′ ∩ r−b′ ⊆ Ar−(a′ ↓b′)]

FTM3a: (∀a′ ∈ S ′)(∀X ′ ∈ C ′(a′))[r−a′ / r−X ′]

FTM3′a: (∀a′ ∈ S ′)(∀X ′ ∈ C ′(a′) ∪D′(a′))(∀V ∈ Pow(S))[r−a′n V → r−X ′ n V ]

FTM3b: (∀a′, b′ ∈ S ′)[a′ ≤′ b′ → r−a′ / r−b′]

FTM4: (∀a′ ∈ S ′)[r−a′ = Ar−a′]

Proof. First suppose that r satisfies the above axioms. To show that r is a balanced

formal topology map, by Proposition 4.1.11 it remains to prove that r satisfies FTM3′.

Let U ′ ∈ Pow(S ′). We must prove that rJ r∗U ′ ⊆ J ′U ′; this can be achieved using

coinduction, by showing that rJ r∗U ′ is a C ′,D′-compatible upper section and a subset

of U ′.

For C ′,D′-compatibility, suppose that a′ ∈ rJ r∗U ′ and X ′ ∈ C ′(a′) ∪ D′(a′).

r−a′n r∗U ′, so by FTM3′a r−X ′ n r∗U ′ and hence X ′ G rJ r∗U ′.

To show that rJ r∗U ′ is an upper section, suppose that a′ ∈ rJ r∗U ′ and

a′ ≤ b′. ara′ for some a ∈ J r∗U ′. By FTM3b we have a / r−b′, so by Com-

patibility J r∗U ′ G r−b′ and hence b′ ∈ rJ r∗U ′.

Finally note that rJ r∗U ′ ⊆ rr∗U ′ ⊆ U ′.

Conversely suppose that r is a balanced formal topology map. From Proposition

4.1.11 we just have to show that FTM3′a is satisfied, so suppose that a′ ∈ S ′, X ′ ∈

C ′(a′) ∪D′(a′), V ∈ Pow(S) and r−a′n V . a′ ∈ rJ V , and by Proposition 3.2.15


rJ V = J ′rJ V . Since J ′rJ V is C ′,D′-compatible we have X ′ G J ′rJ V and hence

X ′ G rJ V , so r−X ′ G J V .

Remark. In the special case where D′(a′) = ∅ for all a′ ∈ S ′, and a / U iff (∀V ∈

Pow(S))[a n V → U n V ], the new axiom FTM3′a is equivalent to FTM3a. These

restrictions on S and S ′, each of which is classically equivalent to AU = (JUC)C ,

will be discussed in Section 7.2.

Postscript

Following the first presentation of this thesis, the author discovered that not every

balanced formal topology can be (co)inductively generated constructively.

Theorem 4.5.8 Let X be a ct-space and let S = ΩBX . If there is a coinductive

definition Φ ⊆ Pow(S) × S such that JU is the largest Φ-progressive subclass of U

for all U ∈ Pow(S), then every ideal point of X is a strong ideal point.

Proof. Any balanced formal topology arising from a ct-space satisfies

a / U ⇔ (∀V ∈ Pow(S))[an V → U n V ] (∗)

For all 〈X, a〉 ∈ Φ, we have (∀V ∈ Pow(S))[a n V → X n V ] since each J V is

Φ-progressive, and hence a / X by (∗).

Now if α is an ideal point of X then for each a ∈ α and X ∈ Pow(S) with

〈X, a〉 ∈ Φ we have a / X and hence α G X. Thus α is Φ-progressive, so α = Jα.

Thus if every balanced formal topology can be (co)inductively generated then

every weakly sober space is sober, and hence the non-constructive principle LPO

holds. This generalises the results of Subsection 7.2.1 and, in the author’s opinion,

places a severe limitation on the usefulness of coinductively defined binary positivity

predicates.

Chapter 5

Metric Spaces and Metric Formal

Topologies

5.1 Metric spaces

The notion of a metric space is well understood in both the classical and constructive

context. In the latter case, metric spaces form a central part of Bishop’s construc-

tivism [15]. A number of topological notions were defined by Bishop only for metric

spaces, as the standard definitions for general topological spaces were found to be less

useful constructively. For example, Bishop defined a metric space to be compact if

and only if it is complete and totally bounded; if the topological notion of compact-

ness in terms of finite subcovers is used, then it is hard to find non-trivial examples

of compact spaces without resorting to Brouwer’s Fan Theorem.

Metric spaces in the constructive context have also been considered by Vickers in

[90, 91, 92], with geometric logic as the foundation. In these papers, Vickers allowed

two important generalisations in his definition of a metric space. Rather than use the

Dedekind reals, which have the property that if r, s ∈ Q, r < s and x ∈ R then either

r < x or x < s, Vickers uses the upper reals1, which do not have this property. We

shall adopt this generalisation here for two reasons. Firstly, it allows a larger class of

1These and other generalised versions of the real numbers have been used under various namesby other constructive mathematicians; see for example [71].

128

CHAPTER 5. METRIC SPACES AND METRIC FORMAL TOPOLOGIES 129

topological spaces to be metrised (for example, the “discrete” topology on any set is

metrisable); secondly, we shall be working with metric formal topologies later, where

the notion of diameter is important, but the diameter of a set of points will not, in

general, be a Dedekind real.

The second generalisation made by Vickers is to drop the symmetry axiom. One

particular advantage of doing this is that the upper and lower powerlocales of the

point-free completion of a (generalised) metric space can then be obtained as gener-

alised metric space completions. However, we shall not consider this generalisation

here. One of the major themes of this thesis is the relationship between point-set and

point-free topological notions, but we have so far been unable to find a satisfactory

point-free version of a generalised metric without the symmetry axiom.

5.1.1 The upper reals

Definition 5.1.1 An upper real is a subset R of Q>0 such that

(∀r ∈ Q>0)[r ∈ R↔ (∃r′ ∈ R)[r′ < r]]

Ru denotes the class of upper reals. We say that R ∈ Ru is:

• finite if ∃r ∈ R

• strongly monotone if (r < s & ¬¬(r ∈ R)) → s ∈ R for all r, s ∈ Q>0

• located or Dedekind if r < s→ (r /∈ R ∨ s ∈ R) for all r, s ∈ Q>0.

The Strong Monotonicity condition defined above is taken from [83], and is of

interest to us because of the connection with the T+i form of the separation properties

(see Propositions 5.1.5 and 5.1.6 below). Note that every Dedekind upper real is

strongly monotone.

Some of the basic properties of the upper reals are discussed in [91]. Given

R,S ∈ Ru, define R + S = r + s | r ∈ R & s ∈ S. For each r ∈ Q≥0, let

r∗ = s ∈ Q>0 | r < s. Write R ≤ S iff S ⊆ R, and R < S if (∃r ∈ Q>0)[R+r∗ ≤ S].

It is easy to show that R < r∗ if and only if r ∈ R. Provided the meaning is clear,

from now on we will write r instead of r∗ for each r ∈ Q≥0.


Lemma 5.1.2 If r ∈ Q>0 and R ∈ Ru, then R ≥ r∗ iff ¬(R < r∗).

Proof. Suppose that R ≥ r∗, that is, R ⊆ r∗. If R < r∗ then there is an s ∈ Q>0

such that r ∈ R+ s∗, that is, r = r′+ s′ for some r′ ∈ R and 0 < s′ < s. Hence r′ < r

and r′ ∈ r∗ — a contradiction.

Now suppose that ¬(R < r∗), and let s ∈ R. Either r < s or r ≥ s. If r < s then

s ∈ r∗, and if r ≥ s then r ∈ R, contradicting ¬(R < r∗). Thus R ⊆ r∗.

As mentioned above, the main advantage that the upper reals have over the

Dedekind reals is that we can define suprema and infima of sets of upper reals. Given

a subset X ⊆ Ru,

supX = r ∈ Q>0 | (∃r′ < r)(∀R ∈ X)[r′ ∈ R]

infX = r ∈ Q>0 | (∃R ∈ X)[r ∈ R]

It is easy to check that supX and infX are upper reals, and that they are the least

upper bound and greatest lower bound of X respectively in Ru.

5.1.2 Ru-metric spaces

Rather than following the usual approach of starting with a metric on a set and then

inducing a topology from the metric, we shall define a metric space by starting with

a ct-space and adding a metric which agrees with the given topology. The reason for

this is that we would like to be able to obtain a metric space from a metric formal

topology (to be defined later), but the points of such a formal topology need not form

a set, and so the usual base consisting of open balls would not be set-indexed.

Definition 5.1.3 A Ru-metric on a class X is a Function d : X × X → Ru such

that

M1: d(x, y) = 0 ↔ x = y

M2: d(x, y) = d(y, x)


M3: d(x, z) ≤ d(x, y) + d(y, z)

for all x, y, z ∈ X. If M1 is weakened to

M1−: d(x, x) = 0 for all x ∈ X

then we call d an Ru-pseudometric on X.

If d(x, y) is finite (resp. strongly monotone, Dedekind) for all x, y ∈ X then we

say that d is finitary (resp. strongly monotone, Dedekind).

Definition 5.1.4 An Ru-(pseudo)metric space is a ct-space X = 〈X, , S〉 equipped

with an Ru-(pseudo)metric d on X such that the topology induced by d agrees with

the given topology on X , that is,

• if x ∈ ext a then Bε(x) ⊆ ext a for some ε ∈ Q>0

• if x ∈ X and ε ∈ Q>0 then x ∈ ext a ⊆ Bε(x) for some a ∈ S.

where Bε(x) = y ∈ X | d(x, y) < ε.

If d is finitary (resp. strongly monotone, Dedekind) then we say that the space X is

finitary (resp. strongly monotone, Dedekind). A metric space is a finitary Dedekind

Ru-metric space.

Given an Ru-pseudometric d on a set X, we can of course define a compatible

topology on X by taking S = X × Q>0 and y 〈x, ε〉 iff d(x, y) < ε. In this case

X = 〈X, , S〉 is a ct-space.

More generally, if d is a Ru-pseudometric on a class X, and X has a dense subset

X0 ⊆ X (that is, (∀x ∈ X)(∀ε ∈ Q>0)(∃y ∈ X)[d(x, y) < ε]), X can be topologised

by taking S = X0 × Q>0 and y 〈x, ε〉 iff d(x, y) < ε. By Proposition 3.2.11, any

open-standard Ru-pseudometric space can be given a topology of this form.

Concerning strongly monotone and Dedekind Ru-metric spaces, we have the fol-

lowing results relating them to the T+i and T ]

i forms of the separation properties; for

the proofs, see [8].

Proposition 5.1.5 Let X be an Ru-metric space.


(i) X is T3.

(ii) If X is strongly monotone then X is T+3 .

(iii) If X is Dedekind then X is T ]3 .

Given a set X with the discrete topology (S = X and x a ↔ x = a), we can

define the discrete Ru-metric d on X by setting

d(x, y) = r ∈ Q>0 | (x = y) ∨ (r > 1)

Proposition 5.1.6 Let X be a set with the discrete metric.

(i) X is strongly monotone iff (∀x, y ∈ X)[¬¬(x = y) → x = y].

(ii) X is Dedekind iff (∀x, y ∈ X)[x = y ∨ ¬(x = y)].

Definition 5.1.7 If X1 and X2 are Ru-metric spaces, a continuous function f : X1 →

X2 is contractive2 if d2(f(x), f(y)) ≤ d1(x, y) for all x, y ∈ X1.

5.2 Metric formal topologies

5.2.1 Diameters

The notion of a diameter on a locale in the classical context was introduced by Pultr

in [66, 67] and explored further in [68], and provides a point-free approach to metric

spaces. Pultr’s definition can be formulated in CZF for open formal topologies, with

some minor changes. However, the diameters defined in this way will usually be

proper classes in CZF. It would seem desirable to be able to define diameters in

terms of sets if possible; this issue will be addressed in Subsection 5.2.2, where we

replace the notion of a diameter with that of a basic diameter. The definition below

is based on that from [68].

2In some of the literature, this is called “non-expansive”


Definition 5.2.1 Let S be a formal topology with a positivity predicate Pos. A Func-

tion ∆ : Pow(S) → Ru is a diameter on S if it satisfies the following axioms:

∆1: ∆(∅) = 0

∆2: U / V → ∆(U) ≤ ∆(V ) for all U, V ∈ Pow(S)

∆3: Pos(U ↓V ) → ∆(U ∪ V ) ≤ ∆(U) + ∆(V ) for all U, V ∈ Pow(S)

∆4: S / a ∈ S |∆(a) < ε for all ε ∈ Q>0

Compatible diameters

Compatibility is defined in [68], and plays two roles. Firstly it means that the diam-

eter gives rise to a metric on Pt(S), rather than just a pseudometric, and secondly it

ensures that the topology on Pt(S) agrees with the topology induced by that metric.

Lemma 5.2.2 Given a diameter ∆ on a formal topology S, define

mcε∆(a) =

⋃U ∈ Pow(S) | (∀V ∈ Pow(S))[Pos(U ↓V ) & ∆(V ) < ε→ V / a]

for each a ∈ S. Then mcε∆(a) is a set for all a ∈ S.

Proof. Given a ∈ S and ε ∈ Q>0, let

W = b ∈ S | (∀c, d ∈ S)[Pos(b↓c) & ∆(c, d) < ε→ d / a]

W is a set by Restricted Separation, and clearly mcε∆(a) ⊆ W .

Now suppose that b ∈ W . Given V ∈ Pow(S) such that Pos(b↓V ) and ∆(V ) < ε,

there must be a c ∈ V such that Pos(b↓ c). For each d ∈ V , ∆(c, d) < ε, so d / a.

Hence V / a. Thus mcε∆(a) = W , so mcε

∆(a) is a set.

Definition 5.2.3 A diameter ∆ on a formal topology S is compatible (with S) if

a /mc∆(a) for all a ∈ S, where mc∆(a) =⋃

ε∈Q>0 mcε∆(a)


Metric diameters

Even classically, Pultr’s definition of Metricity in [68] cannot hold for metric locales

arising from unbounded metric spaces, such as the space of real numbers. It is

possible that the intended meaning was as follows, expressed in the language of

formal topology:

∆(U) = sup∆(V ∪W ) |V,W / U & ∆(V ),∆(W ) < ε (∗)

for all U ∈ Pow(S) and ε ∈ Q>0. This is the definition used more recently by Ba-

naschewski and Pultr in [12], and is classically equivalent to Pultr’s earlier definition

of Metricity from [68] in the case where ∆(U) is finite for all U ∈ Pow(S).

The property (∗) is classically equivalent to the following3, which we shall take as

our constructive definition of metricity.

Definition 5.2.4 A diameter ∆ on a formal topology S is metric if, given U ∈

Pow(S) and r′ < r ∈ Q>0,

(∀a, b ∈ U)[∆(a, b) < r′] → ∆(U) < r

Proposition 5.2.5 (ZF) ∆ is metric if and only if (∗) holds for all U ∈ Pow(S)

and ε ∈ Q>0.

Proof. Suppose that ∆ is metric. Given U ∈ Pow(S) and ε ∈ Q>0, suppose that

sup∆(V ∪W ) |V,W / U & ∆(V ),∆(W ) < ε < r

for some r ∈ Q>0, i.e. there is an r′ < r such that ∆(V ∪W ) < r′ for all V,W / U

such that ∆(V ),∆(W ) < ∆(U). We have to show that ∆(U) < r. Let U ′ = U ↓a ∈

S |∆(a) < ε. U =S U′, so ∆(U) = ∆(U ′).

If a, b ∈ U ′ then a, b / U and ∆(a),∆(b) < ε, so ∆(a, b) < r′. Hence

∆(U ′) < r by Metricity.

3Our definition of metricity can be expressed in the language of locales as d(∨

U) = supd(a ∨b) | a, b ∈ U.


Conversely suppose that (∗) holds. Let U ∈ Pow(S) and r′ < r ∈ Q>0, and

suppose that ∆(a, b) < r′ for all a, b ∈ U . We have to show that ∆(U) < r.

Choose ε ∈ Q>0 such that ε < 13(r − r′) and ε < r′. Suppose that V,W ⊆ AU

and ∆(V ),∆(W ) < ε.

First consider the case where V and W are both positive, so there exist a ∈ V and

b ∈ W such that Pos a and Pos b. Then ∆(V ∪W ) ≤ ∆(V ) + ∆(a, b) + ∆(W ) <

r′ + 2ε.

If V is not positive, then ∆(V ∪W ) = ∆(W ) < ε < r′. Similarly if W is not

positive then ∆(V ∪W ) < r′.

So in each case, ∆(V ∪W ) < r − ε; hence ∆(U) < r by (∗).

5.2.2 Basic diameters

As the diameters defined above are Functions on a proper class, each diameter will

itself be a proper class. This has the potential to cause some difficulties when working

in CZF, and it is likely to make it more difficult to formulate the definition in type

theory.

One solution to this problem, used by Curi in [34, 29, 33], is to define only the

diameters of basic formal opens, i.e. elements of S, rather than defining the diameters

of all subsets of S. Such a diameter is called an elementary diameter. The “shortest

distance” between two basic opens a and b can then be defined in terms of the

lengths of chains of basic opens from a to b. Each elementary diameter is a set,

but this approach has the drawback that defining distances using chains is rather

cumbersome, and an elementary diameter obtained from a diameter does not provide

enough information to recover the original diameter.

The notion of a metric diameter defined above is important because the diameter

of a formal open U ∈ Pow(S) depends only on the diameters ∆(a, b) of pairs

of basic opens. This motivates the following definition, which we shall use for our

predicative point-free approach to metric spaces.


Definition 5.2.6 If S is an open formal topology then a function ν : S × S → Ru is

a basic diameter on S if:

ν1: ν(a, b) = ν(b, a) for all a, b ∈ S

ν2: ν(a, a) ≤ ν(a, b) for all a, b ∈ S

ν3: Pos b→ ν(a, c) ≤ ν(a, b) + ν(b, c) for all a, b, c ∈ S

ν4: S / a ∈ S | ν(a, a) < ε for all ε ∈ Q>0

ν5: supν(a, b) | a, b ∈ U = supν(a, b) | a, b ∈ AU for all U ∈ Pow(S)

The intended meaning is that ν(a, b) represents the diameter of the union of the

basic opens a and b. The axiom ν5 is equivalent to the following: If r′ < r ∈ Q>0,

U ∈ Pow(S) and (∀a, b ∈ U)[ν(a, b) < r′], then (∀a, b ∈ AU)[ν(a, b) < r].

Note that if a′/a and b′/b, then by ν1 and ν2 we have (∀a′′, b′′ ∈ a, b)[ν(a′′, b′′) ≤

ν(a, b)], so ν(a′, b′) ≤ ν(a, b) by ν5.

Definition 5.2.7 Given b ∈ S and ε ∈ Q>0, let

mcεν(b) = a ∈ S | (∀a′ / a)(∀b′ ∈ S)[Pos a′ & ν(a′, b′) < ε→ b′ / b]

and let mcν(b) =⋃

ε∈Q>0 mcεν(b). The basic diameter ν is compatible (with S) if

b /mcν(b) for all b ∈ S.

A metric formal topology is an open formal topology S with a compatible basic

diameter ν.

There is a one-one correspondence between metric diameters and basic diameters,

established in the following propositions.

Proposition 5.2.8 Given a metric diameter ∆ on an open formal topology S, define

ν∆ : S × S → Pow(S) by

ν∆(a, b) = ∆(a, b)

for all a, b ∈ S. Then ν∆ is a basic diameter on S. If ∆ is compatible with S, then

ν is compatible.


Proof. ν1 and ν2 are trivial.

ν3: Let a, b, c ∈ S and Pos b. Then Pos(a, b↓b, c), so by ∆1 and ∆3,

∆(a, c) ≤ ∆(a, b, c) ≤ ∆(a, b) + ∆(b, c)

and hence ν∆(a, c) ≤ ν∆(a, b) + ν∆(b, c).

ν4 trivially follows from ∆4.

ν5: Suppose that r, r′ ∈ Q>0, r′ < r, U ∈ Pow(S) and ν∆(a, b) < r′ for all a, b ∈ U .

Then ∆(U) < r by Metricity, so if a, b ∈ AU then ∆(a, b) < r by ∆2.

Compatibility: It suffices to show that mcε∆(a) ⊆ mcε

ν∆(a) for each a ∈ S and

ε ∈ Q>0. Suppose that U ∈ Pow(S) and (∀V ∈ Pow(S))[Pos(U ↓V ) & ∆(V ) <

ε→ V / a]. We have to show that U ⊆ mcεν∆

(a).

Let b ∈ U . Given a′ ∈ S, b′ / b with Pos b′ and ν∆(a′, b′) < ε, let V = a′, b′.

Then Pos(U ↓V ) and ∆(V ) < ε, so V /a and hence a′/a. Thus b ∈ mcεν∆

(a).

Proposition 5.2.9 Given a basic diameter ν on an open formal topology S, define

∆ν : Pow(S) → Ru by

∆ν(U) = supν(a, b) | a, b ∈ U

= r ∈ Q>0 | (∃r′ < r)(∀a, b ∈ U)[ν(a, b) < r]

= r ∈ Q>0 | (∃r′ < r)(∀a, b ∈ AU)[ν(a, b) < r]

for all U ∈ Pow(S). Then ∆ν is a metric diameter on S, and if ν is compatible with

S then ∆ν is compatible.

Proof. ∆1 and ∆2 are trivial.

∆3: Suppose that U, V ∈ Pow(S) and Pos(U ↓V ), that is, there is a c ∈ U ↓V such

that Pos c. Let r ∈ ∆ν(U) and s ∈ ∆ν(V ). There exist r′ < r and s′ < s such

that ν(a, b) < r′ for all a, b ∈ U , and ν(a, b) < s′ for all a, b ∈ V . To show that

r + s ∈ ∆(U ∪ V ) we will prove that ν(a, b) < r′ + s′ for all a, b ∈ U ∪ V .


If a ∈ U and b ∈ V , choose a′ ∈ U and b′ ∈ V such that c ≤ a′ and c ≤ b′.

Then ν(a, b) ≤ ν(a, c) + ν(c, b) ≤ ν(a, a′) + ν(b, b′) < r′ + s′. The case where

a ∈ V and b ∈ U is similar.

If a, b ∈ U then ν(a, b) < r′ ≤ r′ + s′. The case where a, b ∈ V is similar.

∆4: This follows from ν4 and the fact that ∆(a) = ν(a, a) for all a ∈ S.

Metricity: Let U ∈ Pow(S), r′ < r ∈ Q>0 and suppose that ∆ν(a, b) < r′ for all

a, b ∈ U . If a, b ∈ U then ν(a, b) ≤ ∆ν(a, b) < r′; thus ∆ν(U) < r.

Compatibility: It suffices to show that mcεν(a) ⊆ mcε

∆ν(a) for all a ∈ S and ε ∈ Q>0.

Let b ∈ mcεν(a) and suppose that V ∈ Pow(S), Pos(b↓V ) and ∆ν(V ) < ε. We

have to show that V / a.

Choose a b′ ∈ b↓V such that Pos b′. Given a′ ∈ V , ν(b′, a′) < ε and hence a′ /a.

Thus V / a.

Proposition 5.2.10 Let S be an open formal topology.

(i) If ∆ is a metric diameter on S then ∆ν∆= ∆.

(ii) If ν is a basic diameter on S then ν∆ν = ν.

Hence then operations ∆ 7→ ν∆ and ν 7→ ∆ν give a one-one correspondence between

metric diameters and basic diameters on S. Under this correspondence, compatible

metric diameters correspond to compatible basic diameters.

Proof. (i) Let ∆ be a metric diameter on S. Given U ∈ Pow(S), by Metricity we

have

∆ν∆(U) = sup∆(a, b) | a, b ∈ U = ∆(U)

(ii) Let ν be a basic diameter on S. Given a, b ∈ S,

ν∆ν (a, b) = ∆ν(a, b) = supν(a, a), ν(a, b), ν(b, a), ν(b, b) = ν(a, b)

since ν(a, a), ν(b, b), ν(b, a) ≤ ν(a, b) by ν1 and ν2.

The above correspondence makes it straightforward to reformulate definitions in-

volving metric diameters to work with basic diameters instead.


5.2.3 Contractive maps

Definition 5.2.11 If S1 and S2 are metric formal topologies with basic diameters ν1

and ν2 respectively, then a formal topology map r : S1 → S2 is contractive iff

ν2(a2, b2) ≤ ν1(a1, b1) + ν2(a2, a2) + ν2(b2, b2)

for all a1, b1 ∈ S1 and a2, b2 ∈ S2 such that Pos1 a1, Pos1 b1, a1ra2 and b1rb2.

The following Proposition relates our definition of contractive maps to that used

by Pultr.

Proposition 5.2.12 A formal topology map r : S1 → S2 between metric formal

topologies is contractive if and only if for every ε ∈ Q>0 and U1 ∈ Pow(S1) with

∆ν1(U1) < ε, there is a U2 ∈ Pow(S2) such that U1 /1 r−U2 and ∆ν2(U2) < ε.

Proof. To improve readability, we shall write ∆i in place of ∆νifor i = 1, 2.

First suppose that r is contractive, ε ∈ Q>0, U1 ∈ Pow(S1) and ∆1(U1) < ε. So

there is an ε′ < ε such that (∀a1, b1 ∈ A1U1)[ν1(a1, b1) < ε′]. Let

U2 = a2 ∈ S2 | ν2(a2, a2) < δ & Pos1(U1 ↓r−a2)

where δ = 13(ε − ε′). For each a2, b2 ∈ U2, there exist a1, b1 ∈↓U1 such that Pos1 a1,

Pos1 b1, a1ra2 and b1rb2, and

ν2(a2, b2) ≤ ν1(a1, b1) + ν2(a2, a2) + ν2(b2, b2)

< ε′ + 2δ < ε

Hence ∆2(U2) < ε. To see that U1 /1 r−U2, note that

U1 /1 (U1 ↓r−a2 ∈ S2 | ν2(a2, a2) < δ) ∩ Pos1 ⊆ r−U2

Conversely let r : S1 → S2 satisfy the right-hand side of the proposition, and

suppose that a1, b1 ∈ Pos1, a2, b2 ∈ S2, a1ra2, b1rb2, ν1(a1, b1) < s, ν2(a2, a2) < ε and

ν2(b2, b2) < δ. We have to show that ν2(a2, b2) < s+ ε+ δ.

Let U1 = a1, b1. ∆1(U1) = ν1(a1, b1) < s, so by our first assumption there

is a subset U2 ∈ Pow(S2) such that a1, b1 / r−U2 and ∆2(U2) < s. It follows from


Proposition 3.2.8 that Pos2(a2 ↓U2) and Pos2(b2 ↓U2), so by two applications of ∆3

we have

ν2(a2, b2) = ∆2(a2, b2)

≤ ∆2(U2 ∪ a2, b2)

≤ ∆2(U2 ∪ a2) + ∆2(b2)

≤ ∆2(U2) + ∆2(a2) + ∆2(b2)

< s+ ε+ δ

5.2.4 Dedekind and finitary metric formal topologies

The notions of Dedekind and finitary Ru-metric spaces were defined in Section 5.1.

It is usual in constructive mathematics (see, for example, [15]) to use a system of real

numbers with the quasi-ordering property: if x, y and z are real numbers and x < y,

then either x < z or z < y. This property is satisfied by the Dedekind reals, but not

by the upper reals.

To obtain a point-free version of a Dedekind Ru-metric space, we cannot simply

use the Dedekind reals instead of the upper reals when defining a basic diameter ν,

because ν(a, b) represents the diameter of a set of points, which will not in general

be a Dedekind real, even if the Ru-metric space in question is Dedekind.

The following definition appears to be what is needed. The relationship between

this and the point-sensitive definition will be explained in Section 5.3.

Definition 5.2.13 A metric formal topology S is Dedekind if, given a, b ∈ Pos and

r, s ∈ Q>0 with r < s and ν(a, a) + ν(b, b) < s− r, either ν(a, b) ≥ r or ν(a, b) < s.

We can also formulate a point-free version of a finitary Ru-metric space:

Definition 5.2.14 A metric formal topology S is finitary if

(∀a, b ∈ Pos)(∃a′, b′ ∈ Pos)[a′ / a & b′ / b & (∃r ∈ Q>0)[ν(a′, b′) < r]]

Classically every uniform locale is (completely) regular — see, for example, [66].

Constructively the most we can prove for metric and uniform formal topologies is


2-regularity, but we can also show that every Dedekind metric formal topology is

3-regular. 1-regularity (or weak regularity) of uniform locales was proved by John-

stone in [49].

Proposition 5.2.15 Every metric formal topology is 2-regular.

Proof. By Compatibility, it suffices to prove that if a ∈ mcν(b) then a is 2-well inside

b. Suppose that ε ∈ Q>0 and a ∈ mcεν(b). We have to show that

S / c ∈ S | Pos(b↓c) → c / a

Let U = c ∈ S | ν(c, c) < ε. Then S / U by ν4, and it follows from a ∈ mcεν(b) that

U ⊆ c ∈ S | Pos(b↓c) → c / a.

Proposition 5.2.16 Every Dedekind metric formal topology is 3-regular.

Proof. Given a ∈ S, let

W = b ∈ Pos | (∃ε ∈ Q>0)[b ∈ mcεν(a) & ν(b, b) < 1

4ε]

By Compatibility, a /mcν(a), and if a′ ∈ mcεν(a) then

a′ / (a′↓b ∈ S & ν(b, b) < 14ε) ∩ Pos ⊆ W

so a / W . We will show that if b ∈ W then b is 3-well inside a, that is, S / a ∪ b∗.

Given b ∈ W , let U = c ∈ Pos | ν(c, c) < 14ε. If c ∈ U then ν(b, b)+ν(c, c) < 1

2ε,

so either ν(b, c) ≥ 12ε or ν(b, c) < ε.

If ν(b, c) ≥ 12ε then ¬(ν(b, c) < 1

2ε), so it follows from ν3 that ¬Pos(b ↓ c), and

hence c ∈ b∗. If ν(b, c) < ε then c / a since b ∈ mcεν(a). In either case c / a ∪ b∗;

thus U / a ∪ b∗. Furthermore S / U , so S / a ∪ b∗ as claimed.

It is not true constructively that every metric formal topology is regular. We shall

prove below that if X is an Ru-metric space then ΩX can be given a compatible basic

diameter. By Proposition 3.6.6 X is R] if and only if ΩX is 3-regular, but we have

already seen that not every Ru-metric space is R] constructively.


5.3 A Galois adjunction

5.3.1 From metric spaces to metric formal topologies

Given an open-standard ct-space X with an Ru-metric d, we can define a basic di-

ameter νd on the formal topology ΩOX by setting

νd(a, b) = supd(x, y) |x, y ∈ ext a ∪ ext b

= r ∈ Q>0 | (∃r′ < r)(∀x, y ∈ ext a ∪ ext b)[d(x, y) < r′]

for all a, b ∈ S. It is straightforward to check that νd satisfies the axioms ν1–ν5 and

compatibility. We do, however, need to check that νd(a, b) is a set for all a, b ∈ S.

This can be done using the fact that every open-standard ct-space has a dense subset

(Proposition 3.2.11).

Proposition 5.3.1 If X is a ct-space with an Ru-metric d, and X has a dense subset

X0, then

νd(a, b) = r ∈ Q>0 | (∃r′ < r)(∀x, y ∈ X0 ∩ (ext a ∪ ext b))[d(x, y) < r′]

which is a set by Restricted Separation.

Proof. The inclusion from left to right is trivial. Conversely suppose that r′ < r ∈

Q>0 and (∀x, y ∈ X0 ∩ (ext a ∪ ext b))[d(x, y) < r′]. Let ε = 13(r − r′).

If x, y ∈ ext a ∪ ext b, then there exist x′, y′ ∈ X0 ∩ (ext a ∪ ext b) such that

d(x, x′) < ε and d(y, y′) < ε. d(x′, y′) < r′, so d(x, y) < r′ + 2ε = r − ε. Thus

r ∈ νd(a, b).

5.3.2 From metric formal topologies to metric spaces

Given an open formal topology S with a compatible basic diameter ν, we can define

an Ru-metric dν on Pt(S) by

dν(α, β) = infν(a, b) | a ∈ α & b ∈ β

= r ∈ Q>0 | (∃a ∈ α)(∃b ∈ β)[ν(a, b) < r]


Proposition 5.3.2 Let ν be a compatible basic diameter on an open formal topology

S, and let d = dν. Then d is an Ru-metric on Pt(S), and the topology induced by d

agrees with that of Pt(S).

M1: Given a formal point α of S, we have to show that d(α, α) = 0. For each

r ∈ Q>0, by FP3 and ν4 there is an a ∈ α such that ν(a, a) < r, and hence

d(α, α) < r.

If α, β ∈ Pt(S) and d(α, β) = 0, we have to show that α = β. Suppose that

a ∈ α. By Compatibility and FP3, there is an a′ ∈ α such that a′ ∈ mcεν(a)

for some ε ∈ Q>0. Since d(α, β) < ε, there exist c ∈ α and b ∈ β such that

ν(c, b) < ε. By FP2 there is a c′ ∈ α such that c′ ≤ a′, c. Pos c′ and ν(c′, b) < ε,

hence b / a, so a ∈ β by FP3. Thus α ⊆ β. Similarly β ⊆ α, so α = β.

M2 is trivial.

M3: Let α, β, γ ∈ Pt(S) and suppose that d(α, β) < r and d(β, γ) < s. We have to

show that d(α, γ) < r + s.

By the definition of dν there exist a ∈ α, b1, b2 ∈ β and c ∈ γ such that

ν(a, b1) < r and ν(b2, c) < s. Choose b ∈ β such that b ≤ b1, b2. Then Pos b, so

ν(a, c) ≤ ν(a, b) + ν(b, c) < r + s as required.

Topologies agree: Suppose that a ∈ α for some α ∈ Pt(S). We have to show that

Bε(α) ⊆ ext a for some ε ∈ Q>0. By Metricity and FP3, there is an a′ ∈ α and

an ε ∈ Q>0 such that a′ ∈ mcεν(a). If β ∈ Bε(α) then there exist a′′ ∈ α and

b ∈ β such that ν(a′′, b) < ε; by FP2 a′′ may be chosen so that a′′ ≤ a′. Since

a′ ∈ mcεν(a), it follows that b / a and hence a ∈ β.

Now suppose that α ∈ Pt(S) and ε ∈ Q>0. By ν4 and FP3 there is an a ∈ α

such that ν(a, a) < ε. Clearly α ∈ ext a ⊆ Bε(α).

We end this section with two lemmas relating Ru-metrics on ct-spaces to compat-

ible basic diameters on formal topologies.


Lemma 5.3.3 If X is an Ru-metric space with Ru-metric d, then dνd(αx, αy) =

d(x, y) for all x, y ∈ X.

Proof. Suppose that d(x, y) < r, so d(x, y) < r′ for some r′ < r. Choose a ∈ αx and

b ∈ αy such that ext a ⊆ Bε(x) and ext b ⊆ Bε(y), where ε = 13(r − r′). It follows

from the triangle inequality that νd(a, b) < r, so dνd(αx, αy) < r.

Conversely suppose that dνd(αx, αy) < r. Then there exist a ∈ αx and b ∈ αy

such that νd(a, b) < r, and so d(x, y) < r.

Lemma 5.3.4 If S is a metric formal topology with basic diameter ν, then νdν (a, b) ≤

ν(a, b) for all a, b ∈ S.

Proof. Suppose that ν(a, b) < r, so ν(a, b) < r′ for some r′ < r. By ν2, ν(a, a) < r′

and ν(b, b) < r′, so if α, β ∈ Pt(S) and α, β G a, b then dν(α, β) < r′. Hence

νdν (a, b) < r.

5.3.3 Categories of spaces and formal topologies

Let MSp be the category of Ru-metric spaces and contractive maps, and let MFTop

be the category of metric formal topologies and contractive formal topology maps. If

we say that X1,X2, . . . are Ru-metric spaces, it will be implied that their Ru-metrics

are named d1, d2, . . . unless otherwise stated. We will use the same convention for

metric formal topologies and their basic diameters.

An Ru-metric space is standard if the underlying ct-space is open-standard. A

metric formal topology is standard if the underlying open formal topology is standard.

Let MSpS be the category of standard Ru-metric spaces and standard contrac-

tive maps. Let MFTopS be the category of standard metric formal topologies and

contractive standard formal topology maps.

For each object X of MSpS, let ΩMX be the open formal topology ΩOX (re-

garding X as a ct-space), equipped with the basic diameter νd. For each object S of

MFTop, let PtM(S) be the ct-space PtO(S) equipped with the Ru-metric dν .


Proposition 5.3.5 Given a standard continuous map f : X1 → X2 between stan-

dard metric formal topologies, if f is contractive then ΩOf : ΩMX1 → ΩMX2 is a

contractive formal topology map.

Proof. Let r = Ωf = 〈a1, a2〉 ∈ S1 × S2 | ext a ⊆ f−1(ext b). Suppose that

a1, b1 ∈ S1, Pos a1, Pos b1, a2, b2 ∈ S2, a1ra2 and b1rb2. Since a1, b1 are positive, there

exist points x ∈ ext a1 and y ∈ ext b1, and hence f(x) ∈ ext a2 and f(y) ∈ ext b2.

If x′ ∈ ext a2 and y′ ∈ ext b2 then

d2(x′, y′) ≤ d2(x

′, f(x)) + d2(f(x), f(y)) + d2(f(y), y′)

≤ d2(x′, f(x)) + d1(x, y) + d2(f(y), y′)

≤ νd2(a2, a2) + νd1(a1, b1) + νd2(b2, b2)

and from this it is easy to show that for any x′, y′ ∈ ext a2 ∪ ext b2,

d2(x′, y′) ≤ νd2(a2, a2) + νd1(a1, b1) + νd2(b2, b2)

so νd2(a2, b2) ≤ νd1(a1, b1) + νd2(a2, a2) + νd2(b2, b2).

Proposition 5.3.6 If r : S1 → S2 is a contractive formal topology map between

metric formal topologies, then PtO(r) : PtM(S1) → PtM(S2) is contractive.

Proof. Let α, β ∈ Pt(S1) and suppose that dν1(α, β) < ε, so dν1(α, β) < ε′ for some

ε′ < ε. Let δ = 12(ε− ε′). There is an a2 ∈ rα and a b2 ∈ rβ such that ν2(a2, a2) < δ

and ν2(b2, b2) < δ. So there exist a1 ∈ α and b1 ∈ β such that a1ra2 and b1rb2, and

since dν1(α, β) < ε′ we may choose a1 and b1 so that ν1(a1, b1) < ε′ (using FP2).

ν2(a2, b2) ≤ ν1(a1, b1) + ν2(a2, a2) + ν2(b2, b2)

< ε′ + δ + δ = ε

so dν2(rα, rβ) < ε.

Thus ΩM and PtM can be extended to functors ΩM : MSpS → MFTop and

PtM : MFTop → MSp respectively be setting ΩMf = ΩOf and PtM(r) = PtO(r)

for each standard contractive map f and each contractive formal topology map r.


We have one final result relating contractive maps and contractive formal topology

maps:

Corollary 5.3.7 If f is a standard continuous map between Ru-metric spaces and

ΩOf is a contractive formal topology map, then f is contractive.

Proof. If f : X1 → X2 is continuous, r = ΩOf and r is contractive, then PtO(r) :

PtM(ΩMX1) → PtM(ΩMX2) is contractive. So for all x, y ∈ X1,

d2(f(x), f(y)) = dνd2(αf(x), αf(y))

= dνd2(PtO(r)(αx),PtO(r)(αy))

≤ dνd1(αx, αy)

= d(x, y)

So f is contractive.

In summary, a continuous function f is contractive if and only if ΩOf is contrac-

tive, and if a formal topology map r is contractive then Pt(r) is contractive.

5.3.4 The Galois adjunction

As usual, the unit of the adjunction will be given by the maps ηX : X → Sob(X ),

where ηX (x) = αx for each Ru-metric space X and each point x ∈ X. If Sob(X ) is

given the metric dνd, then it follows from Lemma 5.3.3 that each ηX is contractive.

Furthermore if X is sober then ηX is an isomorphism of Ru-metric spaces.

Theorem 5.3.8 ΩM and PtM restrict to functors ΩMS : MSpS → MFTopS and

PtMS : MFTopS → MSpS respectively, and η is the unit of a Galois adjunction

ΩMS a PtMS.

Proof. From the above observations and the proof of Theorem 3.2.12, it remains to

show that if f : X1 → PtM(S2) is a contractive map then

rf = 〈a1, a2〉 ∈ S1 × S2 | ext a1 ⊆ f−1(ext a2)


is contractive. Suppose that a1, b1 ∈ S1, a2, b2 ∈ S2, Pos1 a1, Pos1 b1, a1rfa2 and

b1rfb2. There exist points x ∈ ext a1 and y ∈ ext b1, and so a2 ∈ f(x) and b2 ∈ f(y).

Suppose that νd1(a1, b1) < r, ν2(a2, a2) < ε and ν2(b2, b2) < δ. Then d1(x, y) < r,

so dν2(f(x), f(y)) < r since f is contractive. There exist a′2 ≤ a2 and b′2 ≤ b2 such

that a′2 ∈ f(x), b′2 ∈ f(y) and ν2(a′2, b

′2) < r. So

ν2(a2, b2) ≤ ν2(a2, a′2) + ν2(a

′2, b

′2) + ν2(b

′2, b2)

< r + ε+ δ

So rf is contractive.

5.3.5 Properties of metric spaces and formal topologies

Proposition 5.3.9 Given an Ru-metric space X ,

(i) if X is finitary then ΩMX is finitary

(ii) if X is Dedekind then ΩMX is Dedekind

Proof. (i) Suppose that X is finitary, that is, d(x, y) is finite for all x, y ∈ X. Given

a, b ∈ Pos, there exist x, y ∈ X and ε, δ ∈ Q>0 such that Bε(x) ⊆ ext a and

Bδ(y) ⊆ ext b. d(x, y) < r for some r ∈ Q>0, and there exist a′, b′ ∈ S such that

x ∈ ext a′ ⊆ Bε(x) and y ∈ ext b′ ⊆ Bδ(y). It follows that νd(a′, b′) < r + ε+ δ,

so dν is finitary.

(ii) Suppose that X is Dedekind. Let r < s in Q>0, a, b ∈ Pos and νd(a, a)+νd(b, b) <

s − r. Since a and b are positive, there exist points x ∈ ext a and y ∈ ext b.

Choose ε ∈ Q>0 such that νd(a, a) + νd(b, b) < s− r− 2ε. Either d(x, y) ≥ r or

d(x, y) < r + ε.

If d(x, y) ≥ r then νd(a, b) ≥ r


If d(x, y) < r + ε then for all x′ ∈ ext a and y′ ∈ ext b,

d(x′, y′) ≤ d(x′, x) + d(x, y) + d(y, y′)

< νd(a, a) + r + ε+ νd(b, b)

< s− r − 2ε+ r + ε

= s− ε

so ν(a, b) < s.

Proposition 5.3.10 Given a metric formal topology S,

(i) if S is finitary then Pt(S) is finitary

(ii) if S is Dedekind then Pt(S) is Dedekind

Proof. (i) Suppose that S is finitary, and let α, β ∈ Pt(S). There exist a ∈ α

and b ∈ β with ν(a, a) < 1 and ν(b, b) < 1. Pos a and Pos b, so there exist

a′, b′ ∈ Pos such that a′ / a, b′ / b and ν(a′, b′) < r for some r ∈ Q>0. Hence by

ν3 we have ν(a, b) < r + 2, so dν(α, β) < r + 2.

(ii) Suppose that S is Dedekind. Given α, β ∈ Pt(S) and r, s ∈ Q>0 with r < s, let

ε = 14(s − r). Choose a ∈ α and b ∈ β with ν(a, a) < ε and ν(b, b) < ε. Then

either ν(a, b) < s or ν(a, b) ≥ s− 2ε.

If ν(a, b) < s then dν(α, β) < s.

If ν(a, b) ≥ s − 2ε, suppose that dν(α, β) < r. Then there exist a′ ∈ α and

b′ ∈ β such that ν(a′, b′) < r, a′ / a and b′ / b.

ν(a, b) ≤ ν(a, a′) + ν(a′, b′) + ν(b′, b)

< ε+ r + ε

= s− 2ε

But this contradicts ν(a, b) ≥ s− 2ε, so ¬(dν(α, β) < r) and hence dν(α, β) ≥

r.


Corollary 5.3.11 If X is a standard Ru-metric space, then

(i) if ΩMX is finitary then X is finitary,

(ii) if ΩMX is Dedekind then X is Dedekind.

Proof. Suppose that ΩMX is finitary (resp. Dedekind). Then PtM(ΩMX ) is finitary

(resp. Dedekind). By Lemma 5.3.3, given x, y ∈ X, d(x, y) = dνd(αx, αy), which is

finite (resp. Dedekind).

To summarise, a standard Ru-metric space X is finitary (resp. Dedekind) if and

only if ΩMX is finitary (resp. Dedekind), and if a metric formal topology S is finitary

or Dedekind then PtM(S) has the same property.

5.4 Metric formal topology completions

In this section we shall show how to construct the completion of a metric formal

topology. This construction is similar to the uniform formal topology completions

which will be covered later in Section 6.5, which in turn are based on Krız’s (classical

impredicative) completion of uniform locales in [53].

Given an Ru-metric space X , its completion can be constructed by taking the space

of formal points of the point-free completion of ΩMX . The point-free completion of

ΩMX is similar to Vickers’ localic completion of generalised metric spaces in [91].

Because of the similarities between metric formal topology completions and the

more general uniform formal topology completions, some of the details will be left

until Section 6.5 to avoid unnecessary repetition.

5.4.1 Adequate bases

In [53], the underlying frame of the completion of a uniform locale is presented by

generators and relations, and there is one generator for every element of the original

frame. Similarly, the completion of a metric or uniform formal topology will be


inductively generated; however the formal opens of the original formal topology S

will usually be a proper class in CZF, so it cannot be used as a base for the completion.

One solution is to use the same base for the completion as for the original formal

topology. For this to be possible, though, we need to make some assumptions about

the original base. To illustrate the problem for the spatial case, consider the Euclidean

metric on X = Q>0, with the topology given by S = Q>0×Q>0 and ext 〈a, b〉 = r ∈

Q>0 | a < r < b.

The Cauchy sequence 1, 12, 1

3, 1

4, . . . should converge to a limit l in the completion

of X, and l must be contained in some basic open set. However, the Cauchy sequence

does not eventually stay inside any basic open set of X. The problem is that X does

not have “enough” basic open sets; if the usual base consisting of open balls Bε(x)

for each x ∈ X and ε ∈ Q>0 is used, then this problem does not arise and the same

base can be used for the completion.

Definition 5.4.1 A metric formal topology S has an adequate base (for the metric

completion) if for all ε ∈ Q>0, there is a δ ∈ Q>0 such that

(∀a ∈ Pos)[ν(a, a) < δ → (∃b ∈ S)[a ∈ mcδν(b) & ν(b, b) < ε]]

This condition is not too restrictive, because given a metric formal topology S we

can construct a richer base by adding the formal opens

Ua,ε = Ab ∈ S | (∃a′ ∈ Pos)[a′ / a & ν(a, b) < ε]

for each a ∈ S and ε ∈ Q>0, as described in Subsection 3.3.4

Note also that if X is an Ru-metric space whose base consists of the open balls

Bε(x) for each x ∈ X (or for each x ∈ X0 for some dense subset X0 ⊆ X), then ΩMX

has an adequate base.

5.4.2 Inductively generating the completion

Given a metric formal topology S with an adequate base, we can define its completion

S to be IG0(〈S,≤′, C〉), where S is the base of S, a ≤′ b iff a / b in S, and


C : S → Pow(Pow(S)) is given by

C(a) = b ∈ Pos | b ∈ mcν(a) & ν(b, b) < ε | ε ∈ Q>0

Let /′ and A′ denote the covering relation and saturation operator belonging to S.

Proposition 5.4.2 A′U ⊆ AU for all U ∈ Pow(S). If a, b ∈ S, then a / b iff

a /′ b.

Proof. ↓′U ⊆ AU and AU is C-closed, so A′U ⊆ AU . a/′ b → a/b is a special

case of this, and a / b→ a /′ b is a result of the preorder ≤′ on S.

It follows that S is a subspace of S. The subspace inclusion is the formal topology

map i = 〈a, b〉 ∈ S × S | a ∈ Ab from S to S.

Proposition 5.4.3 S is a strongly dense subspace of S.

Proof. We have to show that Pos satisfies positivity for S, so that it is a positivity

predicate for S.

Given a ∈ S, a /′ U where U = b ∈ Pos | b ∈ mcν(a). If b ∈ U then b / a and

Pos a, and hence b / a ∩ Pos. Thus a / a ∩ Pos.

Proposition 5.4.4 ν is a compatible basic diameter on S, and the sets mcν(a) are

equal for S and S.

Proof. ν1, ν2 and ν3 are clearly true in S because they do not refer to /′. ν4 follows

from the inductive definition of S, and ν5 is true since A′U ⊆ AU .

The definition of mcν only uses / and /′ with a singleton on the right-hand side,

so mcν(a) is the same for both formal topologies. Compatibility then follows from

this and the inductive definition of C(a).

A metric formal topology is complete if AU = A′U for all U ∈ Pow(S). Since the

definition of A′ only uses /′ with a singleton on the right-hand side, together with mc

and Pos which are the same in S as in S, the completion of a metric formal topology

is always complete.


5.4.3 The universal property

The aim of this section is to prove the following:

Theorem 5.4.5 Given a contractive formal topology map r : S1 → S2 between metric

formal topologies with adequate bases, there is a unique formal topology map r : S1 →

S2 such that the square below commutes

S1⊂

i1 - S1

S2

r

?⊂

i2 - S2

r

?

where i1 and i2 are the subspace inclusions of S1 and S2 into their completions.

Furthermore the map r is contractive.

It follows that we can define a functor from MFTop to the full subcategory of

complete metric formal topologies by S 7→ S on objects and r 7→ r on morphisms.

Functoriality is a consequence of the uniqueness of r.

We shall begin by proving the existence of r. Given r : S1 → S2, let

r = 〈a1, a2〉 ∈ S1 × S2 | a1 /′1 r

−(mcν2(a2))

Lemma 5.4.6 If r : S1 → S2 is contractive and S1 and S2 have adequate bases, then

r is a contractive formal topology map from S1 to S2.

Proof. The fact that r is a formal topology map is a consequence of the corresponding

result, Lemma 6.5.10, for uniform formal topologies, together with Proposition 6.4.2.

To show that r is contractive, suppose that a1, b1 ∈ Pos1, a2, b2 ∈ S2, a1ra2 and

b1rb2. Then a1ra′2 and b1rb

′2 for some a′2 ∈ mcν2(a2) and b′2 ∈ mcν2(b2), so a1ra2 and

b1rb2. So

ν2(a2, b2) ≤ ν1(a1, b1) + ν2(a2, a2) + ν2(b2, b2)

since r is contractive.

Lemma 5.4.7 If r is defined as above, then the square in Theorem 5.4.5 commutes.


Proof. This is a special case of Lemma 6.5.11 for uniform formal topologies.

In [53], using classical logic, the uniqueness of r was deduced from the fact that the

dense morphisms between regular locales are precisely the epimorphisms in the full

subcategory of Loc whose objects are the regular locales. However, constructively it

cannot be shown that every metric formal topology is 3-regular: if X is a standard

Ru-metric space, then ΩMX is 3-regular if and only if X is R], by Proposition 3.6.6,

but if X is the discrete Ru-metric space a set X then X is only R] if equality on X

is decidable.

We have been unable to substitute weak regularity for regularity in Krız’s proof,

but 2-regularity is strong enough, and is satisfied by all Ru-metric spaces.

Lemma 5.4.8 Given a diagram

S1

r- S2

s-

t- S3

of maps between open formal topologies where S3 is 2-regular and r is strongly dense

(i.e. Pos2 a2 → Pos1(r−a2) for all a2 ∈ S2), if s r = t r then s = t.

Proof. We will begin by proving that if Pos2(s−b3 ↓2 t

−c3) then Pos3(b3 ↓3 c3),

for b3, c3 ∈ S3. So suppose that a2 ∈ s−b3↓2 t−c3 and Pos2 a2. Since r is strongly

dense, there is a positive a1 ∈ r−a2, so

a1 /1 (r−s−b3)↓1 (r−t−c3)

=S1 (r−s−b3)↓1 (r−s−c3)

=S1 r−s−(b3 ↓3 c3)

so Pos3(b3 ↓3 c3) as claimed.

Now we will show that s−a3 /2 t−a3. Since S3 is 2-regular, it suffices to prove

that if b3 is 2-well inside a3 then s−b3 /2 t−a3. If b3 is 2-well inside a3, then

s−b3 /2 s−b3↓2 t−S3

/2 s−b3↓2 t−c3 ∈ S3 | Pos3(b3 ↓3 c3) → c3 /3 a3

/2 s−b3↓2 t−c3 ∈ S3 | Pos2(s

−b3↓2 t−c3) → c3 /3 a3


Suppose that a2 is in the right-hand side of the last equation above, and that Pos2 a2.

Then a2 ∈ t−c3 for some c3 ∈ S3 such that Pos2(s−b3↓2 t

−c3) → c3 /3 a3.

a2 ∈ s−b3↓2 t−c3, so Pos2(s

−b3↓2 t−c3) and hence c3/3a3. So a2/2 t

−a3.

This completes the proof that s−a3 /2 t−a3; by symmetry t−a3 /2 s

−a3 so

s = t.

The inclusion i2 is strongly dense, and every metric formal topology is 2-regular

by Proposition 5.2.15, so r is the unique formal topology map which makes the square

in Theorem 5.4.5 commute. This completes the proof of Theorem 5.4.5.

5.4.4 The points of metric formal topology completions

The point-free completion gives rise to a completion operation on standard Ru-metric

spaces, by setting X = PtM(ΩMX ). In this section we shall show that X is the com-

pletion of X (using a suitable constructive notion of completeness). The inductively

generated formal topology ΩMX can also be regarded as a “point-free completion”

of the Ru-metric space X , and is similar to Vickers’ localic completion of generalised

metric spaces [91].

Cauchy points and complete Ru-metric spaces

Let X be a standard Ru-metric space, with metric d, such that ΩMX has an adequate

base for the metric completion. A Cauchy point of X is a formal point of ΩMX . It

is easy to show that α ∈ Pow(S) is a Cauchy point if and only if

(i) (∀ε ∈ Q>0)(∃a ∈ α)(∀x, y ∈ ext a)[d(x, y) < ε]

(ii) (∀a, b ∈ α)(∃c ∈ α)[c ∈ mc(a) ∩mc(b)]

(iii) (∀a ∈ α)(∀b ∈ S)[ext a ⊆ ext b→ b ∈ α]

(iv) (∀a ∈ α)[∃x ∈ ext a]

where mc(a) = mcνd(a) = b ∈ S | (∃ε ∈ Q>0)(∀x ∈ ext b)[Bε(x) ⊆ ext a].


Definition 5.4.9 A Cauchy point α ⊆ S converges if α = αx for some x ∈ X. X is

complete if every Cauchy point converges.

Given any point x ∈ X, αx is a formal point of ΩX , which is a subspace of ΩX ,

so αx is a formal point of ΩX , that is, a Cauchy point of X .

X is complete if and only if the map x 7→ αx is a surjection from X onto X . By

Lemma 5.3.3 and the results of Subsection 5.4.2 we have d(αx, αy) = d(x, y) for all

x, y ∈ X, where d is the Ru-metric of X . It follows that X is complete if and only if

x 7→ αx is an isomorphism between X and X in the category of Ru-metric spaces.

Since every Ru-metric space is T1, if α and β are Cauchy points and α ⊆ β then

α = β.

Cauchy points versus Cauchy sequences

Proposition 5.4.10 If ξ = (xn)n∈N is a Cauchy sequence in X and ΩMX has an

adequate base, then

αξ = a ∈ S | (∃a′ ∈ mc(a))(∃N ∈ N)(∀n ≥ N)[xn ∈ ext a′]

is a Cauchy point of X .

Proof. (iii) and (iv) are trivial. For (i), suppose that ε ∈ Q>0. Since ΩX has an

adequate base, there is a δ ∈ Q>0 such that

(∀a ∈ Pos)[νd(a, a) < δ → (∃b ∈ S)[a ∈ mcδνd

(b) & νd(b, b) < ε]]

Choose N ∈ N large enough so that d(xn, xm) < 13δ for all n,m ≥ N , and choose

a ∈ S such that xN ∈ ext a ⊆ Bδ/3(xN). Then xn ∈ ext a for all n ≥ N , and

furthermore νd(a, a) < δ so there is a b ∈ S such that a ∈ mc(b) and νd(b, b) < ε.

Thus b ∈ αξ.

For (ii), suppose that a, b ∈ α, so there exist a′ ∈ mc(a), b′ ∈ mc(b) and N ∈ N

such that xn ∈ ext a′ ∩ ext b′ for all n ≥ N . There is an ε ∈ Q>0 such that (∀x ∈

ext a′)[Bε(x) ⊆ ext a] and (∀x ∈ ext b′)[Bε(x) ⊆ ext b]. By (i) there is a c ∈ α whose


diameter is less than 12ε. If n is sufficiently large then xn ∈ ext a′ ∩ ext b′ ∩ ext c,

so it follows from the triangle inequality that if x ∈ ext c then Bε/2(x) ⊆ Bε(xn) ⊆

ext a ∩ ext b. Hence c ∈ mc(a) ∩mc(b).

Proposition 5.4.11 If ξ = (xn)n∈N and ζ = (yn)n∈N are Cauchy sequences, then

αξ = αζ if and only if

(∀ε ∈ Q>0)(∃N ∈ N)(∀n ≥ N)[d(xn, yn) < ε] (∗)

Proof. Suppose that (∗) holds, and let a ∈ αξ. There is an a′ ∈ S such that (xn)n∈N

eventually stays inside ext a′, and (∀x ∈ ext a′)[Bε(x) ⊆ ext a] for some ε ∈ Q>0.

Choose b ∈ αζ with diameter less than 12ε. It follows from (∗) that there exist

x ∈ ext a′ and y ∈ ext b with d(x, y) < 12ε. So ext b ⊆ Bε/2(y) ⊆ Bε(x) ⊆ ext a, and

hence a ∈ αζ . Thus αξ ⊆ αζ ; by symmetry αζ ⊆ αξ.

Conversely suppose that αξ = αζ . Given ε ∈ Q>0, choose a ∈ αξ with diameter

less than ε. Both Cauchy sequences eventually stay inside ext a, so (∗) holds.

Proposition 5.4.12 Given a Cauchy sequence ξ = (xn)n∈N, αξ converges to x ∈ X

if and only if ξ converges to x.

Proof. Suppose that αξ converges to x, that is, αξ = αx. Given ε ∈ Q>0, choose

a ∈ αx with with diameter less than ε. Since a ∈ αξ, there is an N ∈ N such that

xn ∈ ext a for all n ≥ N , so d(x, xn) < ε for all n ≥ N .

Conversely suppose that ξ converges to x. Given a ∈ αx, choose b ∈ αx such that

b ∈ mc(a). Since ξ converges to x there is an N ∈ N such that xn ∈ ext b for all

n ≥ N , so a ∈ αξ. Thus αx ⊆ αξ, so αξ converges to x.

It follows from the above results that if X is complete then every Cauchy sequence

converges. It is easy to show that, in the presence of Countable Choice, if α is a

Cauchy point then there is a Cauchy sequence ξ such that α = αξ. Thus (assuming

Countable Choice) if every Cauchy sequence converges then X is complete.


Completeness of PtM(S)

In [11], Banaschewski and Pultr proved (classically) that the “Cauchy points” of a

metric locale form a complete metric space, in the sense that every Cauchy sequence

converges. This proof can be adapted to show that, in CZF, every Cauchy sequence

in PtM(S) converges, if S is a metric formal topology. It is slightly harder to show

that PtM(S) is complete, as defined above using Cauchy points, and we have left the

details until Chapter 6 to avoid repetition.

Theorem 5.4.13 If S is a complete metric formal topology, then PtM(S) is a com-

plete Ru-metric space.

Proof. This is a special case of the corresponding result (Theorem 6.5.18) for uniform

formal topologies.

Corollary 5.4.14 If X is a standard Ru-metric space, then X = PtM(ΩX ) is com-

plete.

5.5 Compactness and total boundedness

In this section we describe a point-free version of the classical result that a metric

space is complete if and only if it is totally bounded. A similar result was proved in

[92]; our proof is very similar but offers a slight generalisation, as we consider point-

free completions of metric formal topologies rather than spaces. The proof in this

section is a special case of Theorem 6.6.3, so the details will be left until Section 6.6.

Definition 5.5.1 An Ru-metric space X is totally bounded if for all ε ∈ Q>0 there

exists a finite subset X0 ⊆ X such that X =⋃

x∈X0Bε(x).

Definition 5.5.2 A metric formal topology S is totally bounded if for all ε ∈

Q>0 there exist U1, . . . , Un ∈ Pow(S) such that S /⋃

i≤n Ui, PosUi and (∀a, b ∈

Ui)[ν(a, b) < ε] for each i ≤ n.


Proposition 5.5.3 A metric formal topology S with an adequate base is totally

bounded if and only if for all ε ∈ Q>0 there exist a1, . . . , an ∈ Pos such that S /

a1, . . . , an and ν(ai, ai) < ε for each i ≤ n.

Proof. The implication from left to right is trivial. For the converse, suppose that

S is totally bounded and let ε ∈ Q>0. Choose δ ∈ Q>0 such that

(∀a ∈ Pos)[ν(a, a) < δ → (∃b ∈ S)[a ∈ mcδν(b) & ν(b, b) < ε]]

Since S is totally bounded there exist U1, . . . , Un ∈ Pow(S) such that S /⋃

i∈I Ui,

PosUi and (∀a, b ∈ Ui)[ν(a, b) < δ] for all i ≤ n. For each i, choose ai ∈ Ui with

Pos ai. ν(ai, ai) < δ, so for each i we can choose a bi ∈ S such that ai ∈ mcδν(bi) and

ν(b, b) < ε. Clearly Ui / ai for all i ≤ n, so S / b1, . . . , bn.

Proposition 5.5.4 A standard Ru-metric space X is totally bounded if and only of

ΩMX is a totally bounded formal topology.

Proof. Suppose that X is totally bounded. Given ε ∈ Q>0, choose x1, . . . , xn ∈ Q>0

such that X =⋃

i≤nBε/3(xi). For each i ≤ n, let Ui = Bε(xi) = a ∈ S | ext a ⊆

Bε(xi). extUi = intBε/3(xi) = Bε/3(xi), so X =⋃

i≤n extUi and hence S /⋃

i≤n Ui.

If x, y ∈ extUi then d(x, y) < 23ε, and it follows that νd(a, b) < ε for all a, b ∈ Ui.

Thus ΩMX is totally bounded.

Conversely suppose that ΩMX is totally bounded. Given ε ∈ Q>0, choose

U1, . . . , Un ∈ Pow(S) such that PosUi and (∀a, b ∈ Ui)[νd(a, b) < ε] for all i ≤ n

and S /⋃

i∈I Ui. Since each Ui is positive, we can choose x1, . . . , xn ∈ X such that

xi ∈ extUi for each i. Given y ∈ X, y ∈ Ui for some i ≤ n and so d(xi, yi) < ε.

Theorem 5.5.5 A metric formal topology S is totally bounded if and only if its

completion S is compact.

Proof. By Proposition 6.6.2 this is a special case of Theorem 6.6.3.


5.6 Metric subspaces

In the point-sensitive case, given a ct-space X with an Ru-metric d, a metric subspace

of X can be defined to be a subclass X ′ ⊆ X, with the Ru-metric given by the

restriction of d to Y .

More generally, metric subspaces of X can be defined by injections into X . Given

a function f : X ′ → X between Ru-metric spaces, we shall say that f is an isometry

if d′(x′, y′) = d(f(x′), f(y′)) for all x′, y′ ∈ X ′. Given such a map, the metric and

the topology on X ′ are uniquely determined by those on X and by the map f , and

furthermore f must be injective. Conversely if f : X ′ → X is an injection, then X ′

can be given the metric and the topology “induced” by f , and this turns f into an

isometry between Ru-metric spaces.

In this section we will define a point-free notion of an isometry, and prove that

it is equivalent to a subspace inclusion in which the domain has the “induced” ba-

sic diameter. Isometries can be easier to work with than subspace inclusions; as

an application we will discuss the subspaces of metric formal topologies and their

completions, and show that a weakly closed subspace of a complete metric formal

topology is complete.

In order to formulate a point-free notion of an isometry, it will be helpful to split

the definition into two parts. Recall that a map f : X1 → X2 between Ru-metric

spaces is contractive if d2(f(x), f(y)) ≤ d1(x, y) for all x, y ∈ X1. We shall call f

expansive if d2(f(x), f(y)) ≥ d1(x, y) for all x, y ∈ X1. Note that this terminology is

not standard. f is an isometry iff it is contractive and expansive.

5.6.1 The basic diameter induced by a subspace inclusion

Definition 5.6.1 If S1 and S2 are formal topologies, ν2 is a basic diameter on S2

and r : S1 → S2 is a formal topology map, the basic diameter on S1 induced by r is

given by

ν1(a1, b1) = supδ(a′1, b′1) | a′1, b′1 ∈ Pos1 & a′1, b′1 / a1, b1


where

δ(a′1, b′1) = infν2(a2, b2) | Pos1(a

′1 ↓r−a2) & Pos1(b

′1 ↓r−b2)

for all a′1, b′1 ∈ S1.

Intuitively δ(a′1, b′1) can be thought of as the shortest distance between the images

in S2 of the two basic opens a′1 and b′1.

Proposition 5.6.2 The basic diameter induced by a formal topology map r : S1 → S2

is a basic diameter on S1.

Proof. ν1 and ν2 are trivial.

ν3: Let a1, b1, c1 ∈ S1, Pos b1, ν1(a1, b1) < r′ < r and ν1(b1, c1) < s′ < s. Choose

ε ∈ Q>0 such that r′ + s′ + ε < r + s. We have to show that ν1(a1, c1) < r + s.

Suppose that a′1, c′1 ∈ Pos1 and a′1, c′1 /1 a1, c1. a′1 /1 a

′1 ↓ a1, c1, so by

Monotonicity there is an a′′1 ∈ a′1 ↓a1, c1 such that Pos a′′1. Similarly there is a

positive c′′1 ∈ c′1 ↓a1, c1. We will show that δ(a′′1, c′′1) < r′ + s′ + ε, from which

it follows that δ(a′1, c′1) < r′ + s′ + ε.

First consider the case where a′′1 ≤ a1 and c′′1 ≤ c1. Since b1 /1 b1 ↓ r−b2 ∈

S2 | ν2(b2, b2) < ε, there is a b′′1 ∈ Pos1 such that b′′1 ≤ b1 and b′′1rb2 for some

b2 ∈ S2 with ν2(b2, b2) < ε. δ(a′′1, b′′1) < r′ and δ(b′′1, c

′′1) < s′, so there exist

a2, b′2, b

′′2, c2 ∈ S2 such that ν2(a2, b

′2) < r′, ν2(b

′′2, c2) < s′ and

Pos1(a′′1 ↓r−a2) Pos1(b

′′1 ↓r−b′2)

Pos1(c′′1 ↓r−c2) Pos1(b

′′1 ↓r−b′′2)

It follows from the two formulae on the right that Pos2(b2 ↓b′2) and Pos2(b2 ↓b′′2),

so we have ν2(a2, c2) ≤ ν2(a2, b′2) + ν2(b2, b2) + ν2(b

′2, c2) < r′ + s′ + ε, and hence

δ(a′′1, c′′1) < r′ + s′ + ε. The case where a′′1 ≤ c1 and c′′1 ≤ a1 is similar.

The other two cases to consider are a′′1, c′′1 ≤ a1 and a′′1, c

′′1 ≤ c1. In the first case

δ(a′′1, c′′1) ≤ ν1(a1, b1) < r′ and in the second case δ(a′′1, c

′′1) ≤ ν1(b1, c1) < s′.


ν4: Given c1 ∈ S1 and c2 ∈ S2 such that c1rc2 and ν2(c2, c2) < ε, we will show that

ν1(c1, c1) ≤ ε.

Suppose that a′1, b′1 ∈ Pos1 and a′1, b′1 /1 c1. Then a′1rc2 and b′1rc2, and it

follows that δ(a′1, b′1) < ε.

ν5: Suppose that U ∈ Pow(S1), r′ < r ∈ Q>0 and (∀a1, b1 ∈ U)[ν1(a1, b1) < r′].

Given a1, b1 ∈ AU , we need to show that ν1(a1, b1) < r.

Let a′1, b′1 ∈ Pos1 and a′1, b′1 /1 a1, b1. a′1 /1 a

′1 ↓U , so by Monotonicity there

is an a′′1 ∈ Pos1 such that a′′1 ≤ a′1 and a′′1 ∈↓U . Similarly there is a b′′1 ∈ Pos1

such that b′′1 ≤ b′1 and b′′1 ∈↓ U . Since (∀a1, b1 ∈ U)[ν1(a1, b1) < r′], we have

δ(a′′1, b′′1) < r′, and hence δ(a′1, b

′1) < r′.

Proposition 5.6.3 If r : S1 → S2 is a subspace inclusion and ν2 is a compatible

basic diameter on S2, then the induced basic diameter ν1 on S1 is compatible.

Proof. Given a1 ∈ S1, since r is a subspace inclusion there is a set U ∈ Pow(S2)

such that a1 =S1 r−U . By the compatibility of ν2, U /

⋃a2∈U mcν2(a2), so we can

prove that ν1 is compatible by showing that r−(⋃

a2∈U mcν2(a2)) ⊆ mcν1(a1).

Suppose that a2 ∈ U , b2 ∈ mcεν2

(a2) and b1rb2. Let η = 12ε. We aim to show that

b1 ∈ mcην1

(a1), so suppose also that a′1 ∈ S1, b′1 ∈ Pos1, b

′1 /1 b1 and ν1(a

′1, b

′1) < η.

There is a subset V ∈ Pow(S2) such that a′1 =S1 r−V . We can choose the set V

so that (∀a′2 ∈ V )[ν(a′2, a′2) < η] (if not, replace V with V ↓a′2 ∈ S2 | ν2(a

′2, a

′2) < η).

To prove that a′1 /1 a1 it suffices to show that given a′′1 ∈ r−V ∩Pos1 we have a′′1 /1 a1.

Suppose that a′′1 ∈ Pos1 and a′′1ra′2 for some a′2 ∈ V . a′′1, b′1 /1 a′1, b′1, so

δ(a′′1, b′1) < η, that is, there exist a′′2, b

′2 ∈ S2 such that ν2(a

′′2, b

′2) < η, Pos1(a

′′1 ↓r−a′′2)

and Pos1(b′1 ↓r−b′2). It follows that Pos2(a

′2 ↓a′′2) and Pos2(b2 ↓b′2). Hence

ν2(b′2, a

′2) ≤ ν2(b

′2, a

′′2) + ν2(a

′2, a

′2) < η + η = ε

so (since Pos2(b2 ↓b′2) and b2 ∈ mcεν2

(a2)) we have a′2 /2 a2. Thus a′′1 /1 r−a2 /1 a1.


5.6.2 Point-free isometries

Recall that, given a formal topology map r : S1 → S2 between metric formal topolo-

gies, r is contractive if and only if

ν2(a2, b2) ≤ ν1(a1, b1) + ν2(a2, a2) + ν2(b2, b2)

whenever a1, b1 ∈ Pos1, a2, b2 ∈ S2, a1ra2 and b1rb2.

Definition 5.6.4 A formal topology map r : S1 → S2 between metric formal topolo-

gies is expansive if ν1(a1, b1) ≤ ν2(a2, b2) whenever a1ra2 and b1rb2.

An isometry is a formal topology map between metric formal topologies which is

both contractive and expansive.

It is easy to see that if f is an expansive map between Ru-metric spaces then Ωf

is an expansive formal topology map.

If r : S1 → S2 is expansive, suppose that α, β ∈ Pt(S1) and dν2(rα, rβ) < ε. Then

there exist a2 ∈ rα and b2 ∈ rβ such that ν2(a2, b2) < ε. We can choose a1 ∈ α and

b1 ∈ β such that a1rb1 and a2rb2. Since r is expansive we have ν1(a1, b1) < ε, so

dν1(α, β) < ε.

If f : X1 → X2 and ΩMf is expansive, then PtM(ΩMf) : Sob(X1) → Sob(X2) is

expansive and so for all x, y ∈ X1,

d2(f(x), f(y)) = dνd2(αf(x), αf(y)) = dνd2

(PtM(ΩMf)(αx), PtM(ΩMf)(αy) )

≤ dνd1(αx, αy) = d1(x, y)

So we have proved:

Proposition 5.6.5 (i) If f : X1 → X2 is a continuous map between standard Ru-

metric spaces, then f is expansive if and only if ΩMf is expansive.

(ii) If r : S1 → S2 is an expansive formal topology map then PtM(r) is expansive.

Corollary 5.6.6 (i) If f : X1 → X2 is a continuous map between standard Ru-metric

spaces, then f is and isometry if and only if ΩMf is an isometry.

(ii) If r : S1 → S2 is an isometry between metric formal topologies then PtM(r)

is an isometry.


Proof. Proposition 5.6.5, and the remark following Corollary 5.3.7.

In the remainder of this section we will prove that the isometries between metric

formal topologies are precisely the subspace inclusions in which the domain has the

induced basic diameter.

Proposition 5.6.7 If r : S1 → S2 is an expansive map between metric formal topolo-

gies then it is a subspace inclusion.

Proof. Given a1 ∈ S1, by Compatibility we have a1 /1 mcν1(a1), so it suffices to show

that mcν1(a1) /1 r−a2 ∈ S2 | r−a2 /1 a1. Suppose that a′1 ∈ mcε

ν1(a1) for some

ε ∈ Q>0. Then

a′1 /1 (a′1 ↓r−b2 ∈ S2 | ν2(b2, b2) < ε) ∩ Pos1

Suppose that Pos a′′1, a′′1 ≤ a′1 and a′′1rb2 for some b2 ∈ S2 with ν2(b2, b2) < ε. We will

show that r−b2 /1 a1 so that a′′1 /1 r−a2 ∈ S2 | r−a2 /1 a1.

Let b1 ∈ S1 and b1rb2. Then ν1(a′′1, b1) ≤ ν2(b2, b2) < ε, so b1 /1 a1 since a′1 ∈

mcεν1

(a1), a′′1 ≤1 a

′1 and Pos1 a

′′1.

Proposition 5.6.8 If r : S1 → S2 is an isometry then the basic diameter on S1 is

that induced by r.

Proof. Let ν1 be the basic diameter on S1, and let ν ′1 be the basic diameter induced

by r. Given a1, b1 ∈ S1, we must show that ν1(a1, b1) = ν ′1(a1, b1).

ν1(a1, b1) ≤ ν ′1(a1, b1): Suppose that r ∈ Q>0 and ν ′1(a1, b1) < r, so there is an r′ < r

such that δ(a′1, b′1) < r′ whenever a′1, b

′1 ∈ Pos1 and a′1, b′1 /1 a1, b1. Let

U = (a1, b1↓c1 ∈ S1 | ν1(c1, c1) < ε) ∩ Pos1

where ε = 13(r − r′). By ν5 it suffices to show that ν1(a

′1, b

′1) < r − ε for all

a′1, b′1 ∈ U .

Given a′1, b′1 ∈ U , δ(a′1, b

′1) < r′ so there exist a2, b2 ∈ S2 such that ν2(a2, b2) <

r′, Pos1(a′1 ↓ r−a2) and Pos2(b

′1 ↓ r−b2). Choose a′′1 ∈ (a′1 ↓ r−a2) and


b′′1 ∈ (b′1 ↓ r−b2) such that Pos1 a′′1 and Pos1 b

′′1. Since r is expansive we have

ν2(a′′1, b

′′1) < r′, and hence

ν1(a′1, b

′1) ≤ ν1(a

′1, a

′′1) + ν1(a

′′1, b

′′1) + ν1(b

′′1, b

′1) < r′ + 2ε = r − ε

ν ′1(a1, b1) ≤ ν1(a1, b1): Suppose that ν1(a1, b1) < r′ < r, and let ε = 13(r − r′). We

will show that δ(a′1, b′1) < r − ε whenever a′1, b

′1 ∈ Pos1 and a′1, b′1 /1 a1, b1.

By Proposition 5.6.7 r is a subspace inclusion, so

a′1 /1 r−a2 ∈ S2 | r−a2 /1 a

′1

/1 r−a2 ∈ S2 | r−a2 /1 a

′1 & ν2(a2, a2) < ε ∩ Pos1

so by Monotonicity there exist a′′1 ∈ Pos1 and a2 ∈ S2 such that a′′1ra2,

r−a2 /1 a′1 and ν2(a2, a2) < ε. Similarly we can find b′′1 ∈ Pos1 and b2 ∈ S2

such that b′′1rb2, r−b2 /1 b

′1 and ν2(b2, b2) < ε.

Pos1(a′1 ↓r−a2) and Pos1(b

′1 ↓r−b2). Furthermore, since r is contractive,

ν2(a2, b2) ≤ ν1(a′′1, b

′′1) + ν2(a2, a2) + ν2(b2, b2)

< ν1(a′1, b

′1) + 2ε < r′ + 2ε = r − ε

so δ(a′1, b′1) < r − ε as claimed.

Proposition 5.6.9 If r : S1 → S2 is a subspace inclusion and S1 has the basic

diameter induced by r, then r is contractive.

Proof. Let a1, b1 ∈ Pos1, a2, b2 ∈ Pos2, a1ra2 and b1rb2. We have to show that

ν2(a2, b2) ≤ ν1(a1, b1) + ν2(a2, a2) + ν2(b2, b2).

Suppose that ν1(a1, b1) < r, ν2(a2, a2) < ε and ν2(b2, b2) < η. δ(a1, b1) < r, so

there exist a′2, b′2 ∈ S2 such that ν2(a

′2, b

′2) < r, Pos1(a1 ↓r−a′2) and Pos1(b1 ↓r−b′2).

It follows that Pos2(a2 ↓a′2) and Pos2(b2 ↓b′2), so

ν2(a2, b2) ≤ ν2(a2, a2) + ν2(a′2, b

′2) + ν2(b2, b2)

< r + ε+ δ


Proposition 5.6.10 If r : S1 → S2 is a subspace inclusion and S1 has the basic

diameter induced by r, then r is expansive.

Proof. Let a1, b1 ∈ S1, a2, b2 ∈ S2, a1ra2 and b1rb2. We have to show that ν1(a1, b1) ≤

ν2(a2, b2), that is, δ(a′1, b′1) ≤ ν2(a2, b2) whenever a′1, b

′1 ∈ Pos1 and a′1, b′1 /1 a1, b1.

Suppose that a′1, b′1 are as described above. Since r is a subspace inclusion, we

have

a′1 /1 r−a2, b2 ↓ r−a′2 ∈ S2 | r−a′2 /1 a′1

/1 r−(a2, b2 ↓ a′2 ∈ S2 | r−a′2 /1 a′1)

so by Monotonicity there is an a′2 ∈ S2 such that a′2 ∈ ↓a2, b2, r−a′2 /1 a′1 and

Pos1(r−a′2). It follows that Pos1(a

′1 ↓r−a′2). Similarly there is a b′2 ∈ S2 such that

b′2 ∈↓a2, b2 and Pos1(b′1 ↓r−b′2). ν2(a

′2, b

′2) ≤ ν2(a2, b2), so δ(a′1, b

′1) ≤ ν2(a2, b2).

If r : S1 → S2 is an isometry (or equivalently if r is a subspace inclusion and S1

has the induced basic diameter), we shall call S1 a metric subspace of S2.

5.6.3 Weakly closed and complete subspaces

Subspaces of metric formal topologies and their completions must be treated with

some care. If we attempt to use the “canonical” forms of subspaces described in

Subsection 3.5.1, in which the formal topology and its subspace share the same base,

then problems can arise when taking the completion. Given a “canonical” subspace

inclusion r : S1 → S2, where S1 and S2 share the same base S, and r−a2 = A1a2

for all a2 ∈ S, it is not necessarily true that the map r : S1 → S2 between the

completions is a “canonical” subspace inclusion.

The following example illustrates the problem. Let X1 = 〈Q ∩ (0, 1), , S〉 and

X2 = 〈Q, , S〉, where S = Q × Q and q 〈a, b〉 iff a < q < b. Let d1 and d2

be the Euclidean metrics on Q ∩ (0, 1) and Q respectively. Now let Si = ΩMXi for

i = 1, 2, and let r be the subspace inclusion from S1 into S2. Taking the completion,

we obtain the map r : S1 → S2 described in the proof of Theorem 5.4.5. If r were


a “canonical” subspace inclusion, then every formal point of S1 would be a formal

point of S2, and hence every Cauchy point of X1 would be a Cauchy point of X2.

But α = 〈p, q〉 ∈ S | p ≤ 0 < q a Cauchy point of X1 but not of X2 — in fact

rα = 〈p, q〉 ∈ S | p < 0 < q.

For this reason we must use the more general form of subspace inclusion, in which

the formal topology and its subspace are not required to share the same base. The

notion of a point-free isometry described earlier in this section provides a convenient

way of proving that a formal topology map between metric formal topologies is a

subspace inclusion and that the domain has the “induced” topology.

Given an isometry S1 → S2 between metric formal topologies, by Theorem 5.4.5

there is a unique formal topology map r : S1 → S2 such that the following square

commutes:

S1⊂

i1 - S1

S2

r

?⊂

i2 - S2

r

?

and the map r is contractive. Since S1 has the same basic diameter as S1 and S2

has the same basic diameter as S2, it is easy to see from the fact that r is expansive

that r is also expansive, so r is an isometry. It can also be shown that the subspace

inclusions i1 and i2 are isometries.

Now consider the case where S2 is complete and S1 is a weakly closed subspace.

S2 is identical to its completion S2 and the subspace inclusion is the identity map, so

the square collapses to

S1

i1 - S1

S2

rr

-

All three maps are isometries, so they are subspace inclusions. By Proposition 5.4.3


the inclusion i1 is strongly dense, so by the definition of weakly closed i1 must be an

isomorphism, and hence S1 and S1 are equal. Thus S1 is complete.

In the classical point-sensitive case, the complete subspaces of a complete metric

space are precisely the closed subspaces. We have so far been unable to prove that

every complete subspace of a metric formal topology is weakly closed.

Chapter 6

Uniform Spaces and Uniform

Formal Topologies

In the past, much of the work on point-free uniform spaces has been carried out the

classical locale-theoretic context. Uniform locales were first studied by Isbell [42],

and more recently by Pultr in [66, 67, 68]. Prior to this thesis, the only known work

on the constructive theory of uniform locales and uniform formal topologies has been

by Johnstone [49] in the impredicative case, and Curi [33, 34] for predicative formal

topology.

On the point-sensitive side, there are a number of different approaches to uniform

spaces. The use of entourages is now the most common, and an introduction to

the subject can be found in [44] (see also [16] and [35]). Another early approach,

introduced by Tukey and defended by Isbell, involved the use of uniform covers. A

brief description of uniform covers and their relationship to entourages is given in [35].

Constructively this relationship is slightly less trivial, and requires an extra property

to be satisfied by the covering uniformities, known as properness. This problem is

discussed in Section 6.1.

Most of the literature on uniform locales, including the impredicative constructive

version in [49], is based on the covering approach to uniformities. A third approach

to uniformities is that of gauges, that is, families of pseudometrics. In the classical

case, this approach is equivalent to the other two (see [52] for the point-sensitive

168

CHAPTER 6. UNIFORM SPACES AND FORMAL TOPOLOGIES 169

case and [67] for the point-free case). The only constructive predicative treatment of

point-free uniformities to date has been [33, 34], where gauges were used instead of

the covering approach. We define uniform formal topologies using covers in Section

6.2, and consider the problem of finding an equivalence between gauges and covering

uniformities in Section 6.4. In the constructive point-sensitive case, the use of gauge

uniformities was preferred by Bishop, and appeared as an exercise in [14] (chapter 4,

problem 17).

The relationship between point-sensitive and point-free uniform spaces will be

examined in Section 6.3. In Sections 6.5 and 6.6 we define the completion of a uniform

formal topology, and show that a uniform formal topology is totally bounded if and

only if its completion is compact, generalising the results of the previous chapter.

6.1 Uniform spaces

6.1.1 Entourage uniformities

Definition 6.1.1 Given a set X, a symmetric entourage of the diagonal is a set

E ⊆ X × X such that ∆ ⊆ E and E = E−1, where ∆ = 〈x, x〉 |x ∈ X and

E−1 = 〈y, x〉 | 〈x, y〉 ∈ E.

Definition 6.1.2 An entourage uniformity on a set X is a set E of symmetric en-

tourages of the diagonal such that

EU1: (∀E1, E2 ∈ E)(∃E ∈ E)[E ⊆ E1 ∩ E2]

EU2: (∀E ∈ E)(∃E ′ ∈ E)[E ′ E ′ ⊆ E]

where F E = 〈x, z〉 | (∃y ∈ X)[〈x, y〉 ∈ E & 〈y, z〉 ∈ F ].

Some of the literature (for example [35]) includes an extra axiom, known as sepa-

ratedness ; we shall treat this as a separate property that may be satisfied by uniform

spaces, and cover it in Subsection 6.1.5.


An entourage uniformity on X determines a neighbourhood system by taking the

neighbourhoods of a point x to be the sets

E(x) = y ∈ X | 〈x, y〉 ∈ E

for each E ∈ E . Thus if X is a small ct-space with an entourage uniformity E , then

the topology on X agrees with that induced by E if and only if

• (∀x ∈ X)(∀a ∈ αx)(∃E ∈ E)[E(x) ⊆ ext a], and

• (∀x ∈ X)(∀E ∈ E)(∃a ∈ αx)[ext a ⊆ E(x)].

Definition 6.1.3 If X and Y have entourage uniformities E and F respectively, a

function f : X → Y is uniformly continuous if

(∀F ∈ F)(∃E ∈ E)(∀〈x, y〉 ∈ E)[〈f(x), f(y)〉 ∈ F ]

If E and E ′ are entourage uniformities on a set X, we say that E ′ is finer than E

(or E is coarser than E ′), if

(∀E ∈ E)(∃E ′ ∈ E ′)[E ′ ⊆ E]

or equivalently if the identity map on X is uniformly continuous when its domain

and codomain are given uniformities E ′ and E respectively.

Two entourage uniformities E and E ′ are equivalent if E is finer than E ′ and E ′ is

finer than E .

6.1.2 Covering uniformities

Given a set X, a cover of X is a set C ∈ Pow(Pow(X)) such that⋃C = X.

If C is a cover of X and Z ⊆ X, let StC(Z) =⋃Y ∈ C |Y G Z. Given two

covers C and C ′, we say that C ′ refines C (written C ′ ≤ C) if

(∀Y ′ ∈ C ′)(∃Y ∈ C)[Y ′ ⊆ Y ]

and C ′ star-refines C (written C ′ <∗ C) if

(∀Y ′ ∈ C ′)(∃Y ∈ C)[StC′(Y ′) ⊆ Y ]


Intuitively each cover C specifies a certain degree of “smallness”, and a set of

points is “small enough” if it is a subset of some Y ∈ C. Refinement, C ′ ≤ C, means

that C ′ expresses a stricter degree of smallness than C. Star-refinement, C ′ <∗ C,

means that if Y is small enough for C ′ then the set of all points that are “within C ′

of Y ” is small enough for C. This is similar to the idea of one entourage being “half

the size” of another.

Definition 6.1.4 A covering uniformity on X is a set C of covers of X such that

CU1: (∀C1, C2 ∈ C)(∃C ∈ C)[C ≤ C1, C2], and

CU2: (∀C ∈ C)(∃C ′ ∈ C)[C ′ <∗ C]

Unfortunately this definition is not sufficient to obtain a correspondence between

entourage uniformities and covering uniformities using intuitionistic logic. To do this,

we shall need an additional property.

Definition 6.1.5 A covering uniformity C on a set X is proper if

(∀C ∈ C)(∃C ′ ∈ C)[C ′ ≤ pC]

where pC = Y ∈ C | ∃x ∈ Y .

Classically every covering uniformity on an inhabited set is proper, because it can

be shown that C ≤ pC for all C ∈ C. Constructively, though, not every covering

uniformity is proper:

Proposition 6.1.6 Let X = 0, 1 and C = C, where C = 0, 1, Z, and

Z = 0 |φ ∪ 1 | ¬φ for some restricted formula φ. If C is proper then ¬φ ∨ ¬¬φ.

Proof. If C is proper then C ≤ pC, so there is a Y ∈ C such that Z ⊆ Y and Y

is inhabited. If Y = 0 then ¬¬φ, and if Y = 1 then ¬φ. If Y = Z then Z is

inhabited, so φ ∨ ¬φ and hence ¬φ ∨ ¬¬φ.

Given a covering uniformity C on a set X, define a neighbourhood system on X

by taking the neighbourhoods of a point x ∈ X to be the sets of the form StC(x)

where C ∈ C. This induces a topology on X.


Given a ct-space X with a covering uniformity C, the topology on X agrees with

the topology induced by C if and only if

• (∀x ∈ X)(∀a ∈ αx)(∃C ∈ C)[StC(x) ⊆ ext a], and

• (∀x ∈ X)(∀C ∈ C)(∃a ∈ αx)[ext a ⊆ StC(x)].

Definition 6.1.7 If X and Y are sets with covering uniformities C and D respec-

tively, a function f : X → Y is uniformly continuous if

(∀D ∈ D)(∃C ∈ C)(∀Y ∈ C)(∃Z ∈ D)[Y ⊆ f−1(Z)]

If C and C ′ are covering uniformities on a set X, we say that C ′ is finer than C (or

C is coarser than C ′) if

(∀C ∈ C)(∃C ′ ∈ C ′)[C ′ ≤ C]

or equivalently if the identity map on X is uniformly continuous when its domain

and codomain are given the covering uniformities C ′ and C respectively.

C and C ′ are equivalent if C is finer than C ′ and C ′ is finer than C.

Equivalence with entourage uniformities

Let X be a set of points. Given an entourage E, we can obtain a cover of X by taking

Cov(E) = E(x) |x ∈ X

Conversely, starting with a cover C, we can define an entourage by

Ent(C) = 〈x, y〉 ∈ X ×X | (∃Y ∈ C)[x, y ∈ Y ]

Proposition 6.1.8 Given an entourage uniformity E, let C(E) = Cov(E) |E ∈ E.

Then C is a proper covering uniformity and the topologies induced by E and C are the

same.

Proof. CU1: Given E1, E2 ∈ E , choose E ∈ E such that E ⊆ E1 ∩ E2. Then

Cov(E) ≤ Cov(Ei) for i = 1, 2.


CU2: Given E ∈ E , choose E ′ ∈ E such that E ′ E ′ E ′ ⊆ E. Given x ∈ X

StCov(E′)(E′(x)) =

⋃E ′(y) | y ∈ X & E ′(y) G E ′(x)

⊆ (E ′ E ′ E ′)(x) ⊆ E(x)

so Cov(E ′) <∗ Cov(E).

Proper: Cov(E) = pCov(E) for all E ∈ E , since x ∈ E(x) for all x ∈ X.

Topologies agree: Given x ∈ X, we have to show that if E ∈ E then there is a

C ∈ C(E) such that StC(x) ⊆ E(x), and that if C ∈ C(E) then there is an

E ∈ E such that E(x) ⊆ StC(x).

For the first part, given E ∈ E , choose E ′ ∈ E such that E ′ E ′ ⊆ E, and let

C = Cov(E ′). Given y ∈ X, if x ∈ E ′(y) then E ′(y) ⊆ (E ′ E ′)(x) ⊆ E(x).

Hence StC(x) ⊆ E(x).

For the second part, given C ∈ C(E), C = Cov(E) for some E ∈ E , and clearly

E(x) ⊆ StC(x).

Proposition 6.1.9 Given a covering uniformity C, let E(C) = Ent(C) |C ∈ C.

Then E(C) is an entourage uniformity and the topologies induced by C and E(C) are

the same.

Proof. EU1: Given C1, C2 ∈ C, choose C ∈ C such that C ≤ C1, C2. Then

Ent(C) ⊆ Ent(C1) ∩ Ent(C2).

EU2: Given C ∈ C, choose C ′ ∈ C such that C ′ <∗ C. If 〈x, y〉, 〈y, z〉 ∈ Ent(C ′)

then there exist Y1, Y2 ∈ C ′ such that x, y ∈ Y1 and y, z ∈ Y2. There is a Y ∈ C

such that StC′(Y1) ⊆ Y . Since Y1 G Y2 we have Y2 ⊆ StC′(Y1). Hence x, z ∈ Y ,

so 〈x, z〉 ∈ Ent(C). Thus Ent(C ′) Ent(C ′) ⊆ Ent(C).

Topologies agree: Given x ∈ X and C ∈ C, it is easy to show that StC(x) =

Ent(C)(x), so the neighbourhood systems giving rise to the two topologies are

the same.


Proposition 6.1.10 Given an entourage uniformity E on a set X, E is equivalent

to E(C(E)).

Proof. For each E ∈ E ,

Ent(Cov(E)) = 〈x, y〉 ∈ X ×X | (∃z ∈ X)[x, y ∈ E(z)]

= E E

Since E ⊆ E E, E is finer than E(C(E)). It follows from EU2 that E(C(E)) is finer

than E .

Proposition 6.1.11 Given a proper covering uniformity C on a set X, C is equiva-

lent to C(E(C)).

Proof. For each C ∈ C,

Cov(Ent(C)) = Ent(C)(x) |x ∈ X

= StC(x) |x ∈ X

To show that C(E(C)) is finer than C, given C ∈ C choose C ′ ∈ C such that

C ′ <∗ C. For each x ∈ X there is a Y ′ ∈ C ′ such that x ∈ Y ′ and hence StC′(x) ⊆

StC′(Y ′) ⊆ Y for some Y ∈ C. Thus Cov(Ent(C ′)) ≤ C.

To show that C is finer than C(E(C)), given C ∈ C choose C ′ ∈ C such that C ′ ≤

pC. If Y ∈ pC then there is a point x ∈ Y and so Y ⊆ StC(x) ∈ Cov(Ent(C)).

Thus C ′ ≤ pC ≤ Cov(Ent(C)).

We have established a one-one correspondence between entourage uniformities

and proper covering uniformities on a set X with a given topology. To construct an

equivalence between the category of entourage uniformities and uniformly continuous

maps, and the category of proper covering uniformities and uniformly continuous

maps, we need to show that the two notions of uniform continuity coincide under

this correspondence.


Proposition 6.1.12 If X and Y are sets with entourage uniformities E and F re-

spectively and f : X → Y is uniformly continuous, then f is also uniformly continu-

ous with respect to the covering uniformities C(E) and C(F) on X and Y .

Proof. Given D ∈ C(F), D = Cov(F ) for some F ∈ F . There is an E ∈ E

such that 〈f(x), f(y)〉 ∈ F for all 〈x, y〉 ∈ E. Let C = Cov(E). For all x ∈ X,

E(x) ⊆ f−1(F (f(x))).

Proposition 6.1.13 If X and Y are sets with covering uniformities C and D respec-

tively and f : X → Y is uniformly continuous, then f is also uniformly continuous

with respect to the entourage uniformities E(C) and E(D) on X and Y .

Proof. Given F ∈ E(D), F = Ent(D) for some D ∈ D. Choose C ∈ C such that

(∀Y ∈ C)(∃Z ∈ D)[Y ⊆ f−1(Z)], and let E = Ent(C). If 〈x, y〉 ∈ E then x, y ∈ Y

for some Y ∈ C, and Y ⊆ f−1(Z) for some Z ∈ D, so f(x), f(y) ∈ Z and hence

〈f(x), f(y)〉 ∈ F .

6.1.3 Open covers

Using covering uniformities instead of entourage uniformities is not, in itself, enough

to allow us to generalise the definition of a uniform space to allow a proper class of

points, as the “elements” of a cover of such a space will themselves be proper classes,

and so cannot be elements of a set.

We will prove that any covering uniformity is equivalent to one in which every

cover contains only open subsets (with respect to the induced topology). This will

allow us to use subsets of Pow(S) to represent the covers, rather than subsets of

Pow(X), and hence obtain a definition of covering uniformities on a ct-space (with

a given topology).

Proposition 6.1.14 Given a covering uniformity C on a set X, let C0 = C0 |C ∈

C, where C0 = intY |Y ∈ C. Then C0 is a covering uniformity and C is equivalent

to C0. Furthermore C0 is proper if and only if C is proper.


Proof. It suffices to show that C is equivalent to C0; the other two claims follow

easily from this fact. If C ∈ C then C0 ≤ C, so C0 is finer than C.

To show that C is finer than C0, given C ∈ C choose C ′ ∈ C such that C ′ <∗ C.

Given Y ′ ∈ C ′ there is a Y ∈ C such that StC′(Y ′) ⊆ Y , and it follows that Y ′ ⊆ intY .

Thus C ′ ≤ C0.

If C = C0, we can replace each cover C ∈ C with a set U ∈ Pow(Pow(S)), where

U = Y |Y ∈ C. The cover C can then be recovered from U since C = extU |U ∈

U. This motivates the following, which we shall take as our definition of a uniform

space.

Definition 6.1.15 An open cover of a ct-space X is a set U ∈ Pow(Pow(S)) such

that X =⋃

U∈U extU .

A uniformity on X is a set U of open covers such that

U1: (∀U1,U2 ∈ U)(∃U ∈ U)[U ≤ U1,U2]

U2: (∀U ∈ U)(∃U ′ ∈ U)[U ′ <∗ U ]

U3: (∀x ∈ X)(∀a ∈ αx)(∃U ∈ U)[StU(x) ⊆ ext a]

where

StU(Y ) = ext⋃U ∈ U |Y G extU

U ≤ V ⇔ (∀U ∈ U)(∃V ∈ V)[extU ⊆ extV ]

U <∗ V ⇔ (∀U ∈ U)(∃V ∈ V)[StU(extU) ⊆ extV ]

The uniformity is proper if (∀U ∈ U)(∃V ∈ U)[V ≤ pU ], where pU = U ∈ U | ∃x ∈

extU.

A (proper) uniform space is a ct-space equipped with a (proper) uniformity.

The axioms U1 and U2 were obtained by rewriting CU1 and CU2 in terms of

open covers. The topology induced by U satisfying U1 and U2 will always be coarser

than the given topology on X , because the covers of U contain only open subclasses

of X. The axiom U3 ensures that the two topologies are the same.


The following result will be useful later.

Lemma 6.1.16 In the presence of U2, the axiom U3 is equivalent to

(∀x ∈ X)(∀a ∈ αx)(∃b ∈ αx)(∃U ∈ U)[StU(ext b) ⊆ ext a]

Proof. =⇒: Assume U3, and let x ∈ X and a ∈ αx. There is a U ∈ U such that

StU(x) ⊆ ext a. Choose V ∈ U with V <∗ U . Since V is a cover, x ∈ ext b for

some V ∈ V and b ∈ V .

StV(ext b) ⊆ StV(extV ) ⊆ extU for some U ∈ U . x ∈ extU , so extU ⊆

StU(x) ⊆ ext a.

⇐=: If b ∈ αx and StU(ext b) ⊆ ext a, then StU(x) ⊆ StU(ext b) ⊆ ext a.

Definition 6.1.17 If X and X ′ are ct-spaces with uniformities U and U′, a function

f : X → Y is uniformly continuous if

(∀U ′ ∈ U′)(∃U ∈ U)(∀U ∈ U)(∃U ′ ∈ U ′)[extU ⊆ f−1(extU ′)]

The uniform ct-spaces and uniformly continuous maps form a category USp.

6.1.4 Uniformities induced by pseudometrics

Given a ct-space X with an Ru-pseudometric d, the most obvious way to induce

a proper uniformity would be to take covers of the form Bε(x) |x ∈ X, where

ε ∈ Q>0. However, these covers may not be sets if X is not a set.

Definition 6.1.18 If d is an Ru-pseudometric on an open-standard ct-space X , let

νd(a, b) = supd(x, y) |x, y ∈ exta, b for all a, b ∈ S, as defined in Subsection

5.3.1. The uniformity induced by d is defined by

U = Uε | ε ∈ Q>0

where Uε = Bε(a) | νd(a, a) < ε & ∃x ∈ ext a

and Bε(a) = b ∈ S | νd(a, b) < ε


By Proposition 5.3.1 (which can be generalised from Ru-metrics to Ru-pseudometrics

without changing the proof), νd(a, b) is a set for all a, b ∈ S, so Bε(a) is a set for all

ε ∈ Q>0 and a ∈ S. Uε is a set for each ε ∈ Q>0, since Pos = a ∈ S | ∃x ∈ ext a is

a set by the definition of open-standard.

Proposition 6.1.19 If X is a small Ru-pseudometric space, then the induced uni-

formity U is equivalent to U′ = U ′ε | ε ∈ Q>0, where U ′ε = Bε(x) |x ∈ X.

Proof. To prove that U refines U′, we will show that Uε ≤ U ′ε for all ε ∈ Q>0. Suppose

that νd(a, a) < ε and there is a point x ∈ ext a. If b ∈ Bε(a) then νd(a, b) < ε, so

ext b ⊆ Bε(x) and hence ext b ⊆ intBε(x) = ext Bε(x).

To prove that U′ refines U, we will show that U ′ε/3 ≤ Uε for each ε ∈ Q>0. Given

x ∈ X, choose a ∈ S such that x ∈ ext a ⊆ Bε/3(x). It follows from the triangle

inequality that νd(a, a) ≤ 23ε < ε, and Pos a since x ∈ ext a. If b ∈ Bε(x) then

νd(a, b) < ε (also by the triangle inequality), so Bε(x) ⊆ Bε(a).

Remark. Impredicatively, one might try to define the uniformity induced by a metric

by using the covers U ′′ε = U ∈ Pow(S) | (∀x, y ∈ extU)[d(x, y) < ε]. There is no

obvious way to predicatively define a uniformity which is equivalent to this. Note

also that if d is the discrete Ru-metric on 0, 1 then U ′′ε = U ⊆ X | ¬(X ⊆ U) for

each ε < 1, and that by a similar argument to Proposition 6.1.6 if this uniformity is

proper then the non-constructive principle wREM is implied.

6.1.5 Separatedness and regularity

Uniform spaces can have a separation property known as separatedness. In some of

the literature (including [16]), separatedness is taken to be part of the definition of a

uniformity. We shall treat it as a property of uniform spaces, separate from the main

definition, as in [44]. Classically every uniform space is regular, and separatedness is

equivalent to the T0 axiom (and hence equivalent to T1, T2 and T3). Constructively

we can define three different forms of separatedness, corresponding to the “ordinary”,

+ and ] forms of the Ti axioms.


Definition 6.1.20 Given a uniform space X with uniformity U, define a binary

Relation ∼U on the points of X by

x ∼U y ⇔ (∀U ∈ U)(∃U ∈ U)[x, y ∈ extU ]

The uniform space is separated if (x ∼U y) ⇒ (x = y) for all x, y ∈ X.

Under the correspondences between the three notions of uniformity, we can define

∼U for a covering uniformity C by setting

x ∼U y ⇔ (∀C ∈ C)(∃Y ∈ C)[x, y ∈ Y ]

and for an entourage uniformity E by

x ∼U y ⇔ 〈x, y〉 ∈⋂E

so that E is separated if and only if⋂E = ∆.

Proposition 6.1.21 Given a uniform space X , the Relation ∼U is equal to ∼i for

i = 0, 1, 2, as defined in Subsection 2.2.1.

Proof. x ∼2 y ⇒ x ∼U y: Suppose that x ∼2 y, and let U ∈ U. Choose V ∈ U such

that V <∗ U . Since V is a cover, there exist V1, V2 ∈ V such that x ∈ V1 and

y ∈ V2.

Using the fact that x ∼2 y, there must be a point z ∈ extV1 ∩ V2, so extV2 ⊆

StV(extV1). There is a U ∈ U such that StV(extV1) ⊆ extU , so x, y ∈ U .

x ∼U y ⇒ x ∼0 y: Suppose that x ∼U y, and let a ∈ αx. By U3 there is a U ∈ U such

that StU(x) ⊆ ext a. Since x ∼U y, there is a U ∈ U such that x, y ∈ extU ,

so y ∈ StU(x) ⊆ ext a and hence a ∈ αy

We have shown that αx ⊆ αy. By symmetry αy ⊆ αx, so αx = αy.

The other required implications, (x ∼0 y) ⇒ (x ∼1 y) and (x ∼1 y) ⇒ (x ∼2 y), hold

for all ct-spaces.


Corollary 6.1.22 For i = 0, 1, 2, a uniform space is separated if and only if it is

Ti.

The + and ] forms of separatedness are defined in a similar way to the stronger

forms of the Ti axioms, by means of an inequality Relation.

Definition 6.1.23 Given a uniform space X , for all x, y ∈ X let

x 6=U y ⇔ (∃U ∈ U)(∀U ∈ U)¬[x, y ∈ extU ]

The space is +-separated if, for all x, y ∈ X,

¬(x 6=U y) ⇒ (x = y)

The space is ]-separated if, for all x, y ∈ X,

(∀a ∈ αx)[y 6=i x ∨ a ∈ αy]

The definition of 6=U can, of course, be adapted for entourage and covering uni-

formities if required, in a similar way to ∼U .

Proposition 6.1.24 Given a uniform space X , the inequality Relation 6=U is equal

to 6=i for i = 0, 1, 2, as defined in Subsection 2.2.1.

Proof. x 6=U y ⇒ x 6=2 y: Suppose that x 6=U y, so there is a U ∈ U such that

(∀U ∈ U)¬[x, y ∈ extU ]. Choose V ∈ U such that V <∗ U . Since V is a cover

there exist V1, V2 ∈ V with x ∈ extV1 and y ∈ extV2.

Suppose that z ∈ extV1 ∩ extV2. Then extV2 ⊆ StV(extV1). There is a U ∈ U

with StV(extV1) ⊆ extU , so x, y ∈ extU , contradicting our earlier assumption.

Thus extV1 ∩ extV2 = ∅, so x 6=2 y.

x 6=0 y ⇒ x 6=U y: Suppose that x 6=0 y, so either there is an a ∈ αx such that a /∈ αy,

or there is a b ∈ αy such that b /∈ αx. Without loss of generality assume the

former.

By U3, StU(x) ⊆ ext a for some U ∈ U. If U ∈ U and x, y ∈ extU then

y ∈ StU(x), contradicting a /∈ αy. So (∀U ∈ U)¬[x, y ∈ extU ].


The implications (x 6=2 y) → (x 6=1 y) and (x 6=1 y) ⇒ (x 6=0 y) are true in any

ct-space.

Corollary 6.1.25 For i = 0, 1, 2, a uniform space is +-separated (resp. ]-separated)

if and only if it is T+i (resp. T ]

i ).

Regularity and the T3 properties

Proposition 6.1.26 Every uniform space is R.

Proof. Given a uniform space X , let x ∈ X and a ∈ αx. By Lemma 6.1.16 there is

a b ∈ αx and a U ∈ U such that StU(ext b) ⊆ ext a.

Suppose that y ∈ cl(ext b). Since U is a cover, there is a U ∈ U such that y ∈ extU .

extU G ext b, so y ∈ StU(ext b) ⊆ ext a. Thus cl(ext b) ⊆ ext a.

It follows from this, together with Corollary 6.1.22, that a uniform space is sep-

arated if and only if it is T3. It is not true constructively that every uniform space

is R+, because every Ru-metric space is uniformisable and by Propostions 5.1.5 and

5.1.6 not every Ru-metric space is R+. It is not clear whether there is any connection

between +- and ]-separability and the T+3 and T ]

3 properties.

6.2 Uniform formal topologies

Our definition of a uniformity on a formal topology will be essentially the same as

the one used by Johnstone in [49], except that for reasons of predicativity rather than

define a uniformity to be a filter on the collection of open covers, we must work with

a base for that filter.

Definition 6.2.1 Given a formal topology S, a cover of S is a set U ∈ Pow(Pow(S))

such that S /⋃U . Let Cov(S) denote the class of covers of S.

Three different notions of star-refinement are considered in [49], where they were

denoted by <∗, <∗1 and <∗

2. <∗ is the strongest of the three, and was found by


Johnstone to be the most satisfactory constructively. <∗2 could only be defined for

open locales, but the other two notions made sense for all locales.

Two out of the three notions, <∗ and <∗1, make sense for all locales, regardless of

whether they are open. However, their definitions both make use of binary products

of locales, which are not available to us predicatively. As mentioned in [49], this can

be avoided if the locale is open. We shall use <∗ in our definition of uniform formal

topologies, and restrict our attention to uniformities on open formal topologies.

Given covers U and V of S, and a subset V ∈ Pow(S), define

StU(V ) =⋃U ∈ U | Pos(U ↓V )

U ≤ V ⇔ (∀U ∈ U)(∃V ∈ V)[U / V ]

U <∗ V ⇔ (∀U ∈ U)(∃V ∈ V)[StU(U) / V ]

We say that U refines V if U ≤ V , and U star-refines V if U <∗ V .

The following lemma will be needed later.

Lemma 6.2.2 If U <∗ V then StU(StU(W )) / StV(W ) for all W ∈ Pow(S).

Proof. Suppose that U ∈ U and Pos(U ↓ StU(W )). There is a U ′ ∈ U such that

Pos(U ↓U ′) and Pos(U ′ ↓W ). U,W ⊆ StU(U ′)/V for some V ∈ V , and Pos(V ↓W ),

so U / V ⊆ StV(W ).

We are now in a position to adapt Johnstone’s definition of (pre)uniformities from

[49] for use with predicative formal topology.

Definition 6.2.3 A preuniformity on an open formal topology S is a set U of covers

satisfying:

U1: (∀U ,V ∈ U)(∃W ∈ U)[W ≤ U ,V ]

U2: (∀U ∈ U)(∃V ∈ U)[V <∗ U ]

We say that U is a uniformity if it also satisfies the following condition:

U3: (∀a ∈ S)[a / uc(a)]


where uc(U) = b ∈ S | (∃U ∈ U)[StU(b) / U ] and uc(a) means uc(a).

A (pre)uniformity U is proper if

U4: (∀U ∈ U)(∃V ∈ U)[V ≤ pU ], where pU = U ∈ U | PosU.

A (proper) uniform formal topology is an open formal topology equipped with a

(proper) uniformity.

Definition 6.2.4 If S and S ′ are open formal topologies with uniformities U and U′

respectively, a formal topology map r : S → S ′ is uniform, or uniformly continuous,

if

(∀U ′ ∈ U′)(∃U ∈ U)(∀U ∈ U)(∃U ′ ∈ U ′)[U / r−U ′]

The uniform formal topologies and uniform formal topology maps form a category,

UFTop.

6.3 Two Galois Adjunctions

6.3.1 From uniform spaces to uniform formal topologies

The relationship between uniform spaces and uniform formal topologies is much sim-

pler than that between metric spaces and metric formal topologies, because the point-

free definition is very similar to the point-sensitive one.

Proposition 6.3.1 Given an open-standard ct-space X and a set U of open covers,

U is a uniformity on X if and only if it is a uniformity on the formal topology ΩOX .

Proof. For U1 and U2, observe that

• extStU(V ) = StU(extV ) for each cover U and each subset V ⊆ S,

• U ≤ V in X if and only if U ≤ V in ΩOX , and

• U <∗ V in X if and only if U <∗ V in ΩOX .


By Lemma 6.1.16 U3 for X is equivalent to

(∀a ∈ S)[ a / b ∈ S | (∃U ∈ U)[StU(ext b) ⊆ ext a] ]

which by the above remarks is equivalent to

(∀a ∈ S)[ a / b ∈ S | (∃U ∈ U)[StU(b) / a] ]

that is, (∀a ∈ S)[a / uc(a)].

It is also easy to see that U is a proper uniformity on X if and only if it is a proper

uniformity on ΩOX .

Proposition 6.3.2 Given a standard continuous map f : X → X ′ between open-

standard uniform spaces, f is uniformly continuous if and only if the formal topology

map ΩOf : ΩOX → ΩOX ′ is uniform.

A uniform space X is standard if the underlying ct-space is open-standard. Let

USpS be the category of standard uniform spaces and standard uniformly continuous

maps.

We can define a functor ΩU : USpS → UFTop as follows. If X is a standard

uniform space, let ΩUX be the formal topology ΩOX with the same uniformity. Given

a standard uniformly continuous map f : X → X ′, let ΩUf = ΩOf .

6.3.2 From uniform formal topologies to uniform spaces

Proposition 6.3.3 If S is a uniform formal topology with uniformity U, then U is

also a uniformity on the ct-space Pt(S).

Proof. For U1 and U2, note that

• if U is a cover of S then it is a cover of Pt(S),

• if U ≤ V in S then U ≤ V in Pt(S),

• StU(extV ) ⊆ extStU(V ) (where ext refers to the space Pt(S)), and hence


• if U <∗ V in S then U <∗ V in Pt(S).

For U3, note that if α ∈ Pt(S) and a ∈ α then by U3 for S and FP3 there exist b ∈ α

and U ∈ U such that StU(b)/a. Hence StU(α) ⊆ StU(ext b) ⊆ extStU(b) ⊆ ext a.

Proposition 6.3.4 If S and S ′ are uniform formal topologies and r : S → S ′ is a

uniform formal topology map, then Pt(r) : Pt(S) → Pt(S ′) is a uniformly continuous

map if Pt(S) and Pt(S ′) are given the same uniformities as S and S ′.

We can define a functor PtU : UFTop → USp. Given a uniform formal topology

S with uniformity U, let PtU(S) be the ct-space Pt(S), with the same uniformity U.

Given a uniform formal topology map r : S → S ′, let PtU(r) = Pt(r).

While it follows from Proposition 3.2.9 that if PtU(S) is proper then so is S, we

have been unable to prove the converse constructively, so PtU cannot be restricted

to a functor from proper uniform formal topologies to proper uniform spaces. This is

our principal reason for not including properness in the point-free and point-sensitive

definitions of a uniformity. If properness were added to these definitions, the functors

described in Section 6.3.4 would have to be used instead.

6.3.3 The Galois adjunction for uniform spaces and formal

topologies

Let UFTopS be the subcategory of UFTop whose objects are uniform formal topolo-

gies which are standard (considered as open formal topologies), and whose morphisms

are standard uniform formal topology maps.

The unit of the Galois adjunction will, again, be given by the maps ηX : X →

Sob(X ), where ηX (x) = αx for each uniform space X and each x ∈ X. If Sob(X ) is

given the same uniformity as X , then ηX is uniformly continuous.

Theorem 6.3.5 ΩU and PtU restrict to functors ΩUS : USpS → UFTopS and

PtUS : UFTopS → USpS respectively, and η is the unit of a Galois adjunction

ΩUS a PtUS.


Proof. From the proof of Theorem 3.2.12, it remains to prove that if f : X1 →

PtU(S2) is uniformly continuous then the formal topology map

rf = 〈a1, a2〉 ∈ S1 × S2 | ext a1 ⊆ f−1(ext a2)

is uniform. Given U2 ∈ U2, choose U1 ∈ U1 such that

(∀U1 ∈ U1)(∃U2 ∈ U2)[extU1 ⊆ f−1(extU2)]

If extU1 ⊆ f−1(extU2), then U1 /1 f−1(extU2) = r−f U2. Hence rf is uniform.

6.3.4 The Galois adjunction for proper uniform spaces and

formal topologies

Let UPSp and UPSpS be the full subcategories of USp and USpS respectively

whose objects are the proper uniform spaces in those categories, and let UPFTop

and UPFTopS be the full subcategories of UFTop and UFTopS respectively whose

objects are proper formal topologies.

Let F be the forgetful functor from UPSpS to USpS. The functor ΩUS F :

UPSpS → UFTopS restricts to a functor ΩUP S : UPSpS → UPFTopS.

Proposition 6.3.6 Given a standard uniform space X with uniformity U, let

U′ = pU | U ∈ U

Then U′ is the coarsest proper uniformity which is finer than U.

Proof. First we have to check that U′ is a proper uniformity. Clearly pU is a cover

for each U ∈ U.

To see that U1 holds, note that if U ≤ V then pU ≤ pV : given U ∈ pU , choose

V ∈ V such that U / V . PosU , so PosV by Monotonicity. Similarly if U <∗ V then

pU <∗ pV , so U′ also satisfies U2.

StpU(V ) = StU(V ) for all U ∈ U and V ∈ Pow(S), so U3 holds.

U4 holds since pU = ppU for all U ∈ U.


U′ is finer than U, because pU ≤ U for all U ∈ U. Now suppose that V is another

proper uniformity which is finer than U. Given U ′ ∈ U′, U ′ = pU for some U ∈ U.

There is a V ∈ V such that V ≤ U , and there is a V ′ ∈ V such that V ′ ≤ pV ≤ pU .

So V is finer than U′.

If X is a standard uniform space, let GX be the uniform space with the same

underlying ct-space X , and the proper uniformity U′ described in the last proposition.

Proposition 6.3.7 If f : X → Y is uniformly continuous, then f is also a uniformly

continuous map GX → GY. Thus G extends to a functor G : USpS → UPSpS.

Proof. Let U and V be the uniformities on X and Y , and let U′ and V′ be the

uniformities on GX and GY .

Given V ′ ∈ V′, V ′ = pV for some V ∈ V, and there is a U ∈ U such that

(∀U ∈ U)(∃V ∈ V)[extU ⊆ f−1(extV )]. If U ∈ pU , then there is a V ∈ V such that

extU ⊆ f−1(extV ), and V ∈ pV since PosU . So f is uniformly continuous w.r.t. U′

and V′.

Proposition 6.3.8 Given a standard proper uniform space X and a standard uni-

form space Y, a function f : X → Y is a uniformly continuous map FX → Y if and

only if it is a uniformly continuous map X → GY.

Proof. X and FX have the same uniformity, and the uniformity on Y is coarser

than the uniformity on GY , so if f : X → GY is uniformly continuous then f is a

uniformly continuous map FX → Y . The converse follows from Proposition 6.3.7,

since the uniformities on X , FX and GFX are equivalent.

It follows that there is an adjunction F a G. The unit of this adjunction is given

by the identity maps η′X : X → GFX for each proper uniform space X , and the

counit is given by the identity maps ε′Y : FGY → Y for each uniform space Y . The

unit is always an isomorphism, and the counit ε′Y is an isomorphism if Y is proper,

so the adjunction is a Galois adjunction.


Composing the two pairs of adjoint functors obtained so far, we can see that there

is a Galois adjunction (ΩUS F ) a (G PtUS). The functors ΩUS F and G PtUS

can be restricted to functors ΩUP S : UPSpS → UPFTopS and PtUP S : UPFTopS →

UPSpS.

6.4 Gauges and metrisation

In this section we begin by showing how to induce a preuniformity from a basic diam-

eter. Every metric formal topology gives rise to a uniform formal topology, so many

of the results in the remainder of this chapter generalise those from Chapter 5. We

also consider the notion of a gauge, that is, a family of basic diameters. This is similar

to the treatment of uniform formal topologies in [33, 34], except we shall use basic

diameters instead of elementary diameters. The point-free use of gauges is analogous

to the point-sensitive approach to uniform spaces using families of pseudometrics (see

[52]), and similar notions have been explored in the classical point-free case by Pultr

[67] and by Picado [65].

Every gauge on a formal topology induces a uniformity. We shall show that given

uniformity satisfying a certain condition (which is needed to avoid axiom of choice)

we can find a gauge which induces it. The proof is similar to that in [67], which in

turn is based on the point-sensitive proof in [52].

6.4.1 Preuniformities induced by basic diameters

Given a (not necessarily compatible) basic diameter ν on a formal topology S, let

U = Uνε | ε ∈ Q>0,

where Uνε = Bν

ε (a) | ν(a, a) < ε & Pos a

and Bνε (a) = b ∈ S | ν(a, b) < ε

for each ε ∈ Q>0. We shall call U the preuniformity induced by ν.

Proposition 6.4.1 U is a proper preuniformity on S.


Proof. First we need to check that Uνε ∈ Cov(S) for each ε ∈ Q>0. But a ∈ Bν

ε (a)

for each positive a ∈ S with ν(a, a) < ε, and S / a ∈ S | Pos a & ν(a, a) < ε, so

S /⋃Uε.

U1: If ε, η ∈ Q>0 and η < ε, then Bνη (a) / Bν

ε (a) for all a ∈ S, and hence Uνη ≤ Uν

ε .

Given any ε, δ ∈ Q>0, let η = minε, δ. Then Uνη ≤ Uν

ε ,Uνδ .

U2: Given ε ∈ Q>0, let δ = 14ε. To show that Uν

δ <∗ Uνε , we will prove that given

a ∈ S with Pos a and ν(a, a) < δ we have StUνδ(Bδ(a)) ⊆ Bε(a).

Suppose that U ∈ Uνδ and Pos(U ↓Bν

δ (a)). Then U = Bνδ (b) for some b ∈ Pos

with ν(b, b) < δ, and there is a c ∈ S such that Pos c and c ∈ U ↓Bνδ (a). Thus

ν(a, c) < δ and ν(c, b) < δ, so ν(a, b) < 2δ. For each d ∈ U , ν(b, d) < δ, so

ν(a, d) < 3δ and hence d ∈ Bνε (a).

Proper: Clearly Uνε = pUν

ε for all ε ∈ Q>0.

Proposition 6.4.2 uc(a) = mcν(a) for all a ∈ S.

Proof. Suppose that b ∈ uc(a). Then StUνε(b)/a for some ε ∈ Q>0. If a′, b′ ∈ S, b′/b,

Pos b′ and ν(a′, b′) < ε, then Pos(Bνε (b′)↓b) and Bν

ε (b′) ∈ Uνε , and so a′ ∈ Bν

ε (b′)/a.

Thus b ∈ mcεν(a).

Now suppose that b ∈ mcεν(a). We will show that StUν

δ(b) / a, where δ = 1

2ε.

Suppose that c ∈ Pos, ν(c, c) < δ and Pos(b ↓Bνδ (c)). There is a positive b′ / b such

that ν(b′, c) < δ. For each d ∈ Bνδ (c), ν(b′, d) ≤ ν(b′, c) + ν(c, d) < 2δ and hence d / a

because b ∈ mcεν(a). Thus Bν

δ (c) / a.

Corollary 6.4.3 U is a uniformity on S if and only if ν is a compatible basic di-

ameter on S.

6.4.2 Metrisation of uniform formal topologies

The aim of this section is to find sufficient conditions under which, for a given pre-

uniformity U, there is a basic diameter ν which induces a preuniformity equivalent

to U. If this is the case, we shall say that U is metrisable. This subject has been


covered in Locale Theory by Pultr in [67] and more recently by Banaschewski and

Pultr in [12], but both papers used classical logic and the axiom of choice.

Classically U is metrisable if and only if it admits a countable base (that is, if U

is equivalent to a uniformity V such that the set V of covers is countable). To avoid

the use of the axiom of choice, we need to strengthen this condition slightly; we shall

prove the following:

Theorem 6.4.4 Given a sequence (Un)n∈N of covers such that U0 = pS, Un = pUn

and Un+1 <∗ Un for all n ∈ N, the preuniformity U = Un |n ∈ N is metrisable.

Let (Un)n∈N be a sequence of covers of S satisfying the conditions of Theorem

6.4.4. Define a function δ : S × S → Ru by

δ(a, b) = inf2−n | (∃U ∈ Un)[a, b / U ]

= r ∈ Q>0 | (∃n ∈ N)(∃U ∈ Un)[a, b / U & 2−n < r]

Given a, b ∈ S, let Ch(a, b) be the set of chains from a to b, that is, lists 〈a0, . . . , an+1〉

of positive elements of S such that a0 / a and an+1 / b.

For each chain 〈a0, . . . , an〉, let `(〈a0, . . . , an+1〉) =∑n

i=0 δ(ai, ai+1), the length of

the chain. Define ν : S × S → Ru by

ν(a, b) = supinf`(C) |C ∈ Ch(a′, b′) | a′, b′ / a, b & a′, b′ ∈ Pos

= r ∈ Q>0 | (∃r′ < r)(∀a′, b′ ∈ Pos)[a′, b′ / a, b

→ (∃C ∈ Ch(a′, b′))[`(C) < r′]]

To prove Theorem 6.4.4, we will show that ν is a basic diameter and that the pre-

uniformity it induces is equivalent to U.

Lemma 6.4.5 ν is a basic diameter on S.

Proof. ν1 and ν2 follow easily from the definition of ν.

ν3: Suppose that a, b, c ∈ S, Pos b, ν(a, b) < r and ν(b, c) < s. There is an ε ∈ Q>0

such that ν(a, b) < r − ε and ν(b, c) < s− ε. Given a′, c′ ∈ Aa, c with Pos a′


and Pos c′, we will show that there is a chain from a′ to c′ whose length is less

than r + s− ε.

By Monotonicity, either Pos(a′ ↓ a) or Pos(a′ ↓ c), and either Pos(c′ ↓ a) or

Pos(c′ ↓ c). If Pos(a′ ↓ a) and Pos(c′ ↓ a), then there is a chain from a′ to c′ of

length less than r− ε. If Pos(a′ ↓c) and Pos(c′ ↓c) then there is a chain from a′

to c′ whose length is less than s − ε. In both cases, the length of the chain is

less than r + s− ε.

Now suppose that Pos(a′ ↓ a) and Pos(c′ ↓ c). Choose n ∈ N large enough so

that 2−n < ε. Since Un is a cover and b is positive, there is a U ∈ Un and a

b′ ∈ (b↓U) such that Pos b′. Clearly δ(b′, b′) < ε.

There exist chains 〈a0, . . . , an+1〉 from a′ to b′ and 〈c0, . . . , cm+1〉 from b′ to c′,

whose lengths are less than r − ε and s − ε respectively. Since an+1 / b′ and

c0 / b′, δ(an+1, c0) ≤ δ(b′, b′) < ε. It follows that

`(〈a0, . . . , an+1, c0, . . . , cm+1〉) ≤ `(〈a0, . . . , an+1〉) + `(〈c0, . . . , cm+1〉) + ε

< (r − ε) + (s− ε) + ε

= r + s− ε

The case where Pos(a′ ↓c) and Pos(c′ ↓a) is similar.

ν4: Given ε ∈ Q>0, choose an n ∈ N such that 2−n < ε. S /⋃Un, and if a ∈ U ∈ Un

then δ(a, a) < 2−n and hence ν(a, a) < 2−n.

ν5: Suppose that U ∈ Pow(S), (∀a, b ∈ U)[ν(a, b) < r′] and r′ < r. We have to show

that ν(a, b) < r for all a, b ∈ AU . It suffices to show that if a′, b′ ∈ AU , Pos a′

and Pos b′ then there is a chain from a′ to b′ whose length is less than r′.

Let a′, b′ ∈ AU both be positive. By Monotonicity there exist a′′, b′′ ∈ U such

that Pos(a′ ↓ a′′) and Pos(b′ ↓ b′′). Choose a′′′, b′′′ ∈ Pos such that a′′′ ∈ a′ ↓ a′′

and b′′′ ∈ b′ ↓ b′′. There is a chain C ∈ Ch(a′′′, b′′′) of length less than r′, and

Ch(a′′′, b′′′) ⊆ Ch(a′, b′).


Lemma 6.4.6 If a1, a2, a3, a4 ∈ S, Pos a2 and Pos a3, then

δ(a1, a4) < 2 maxδ(a1, a2), δ(a2, a3), δ(a3, a4)

Proof. Suppose that r ∈ Q>0 and δ(ai, ai+1) < r for i = 1, 2, 3. There exist n ∈ N

and U1, U2, U3 ∈ Un such that 2−n < r, and ai, ai+1 / Ui for i = 1, 2, 3.

If n ≥ 1, then Un <∗ Un−1 and so there is a V ∈ Un−1 such that StUn(U2) / V . So

a1, a4 / U1 ∪ U3 ⊆ StUn(U2) / V and hence δ(a1, a4) ≤ 2−n+1 < 2r.

If n = 0, then a1, a4 ∈ S ∈ U0 and therefore δ(a1, a4) ≤ 1 < r < 2r.

Lemma 6.4.7 Each chain 〈a0, . . . , an+1〉 satisfies δ(a0, an+1) ≤ 2`(〈a0, . . . , an+1〉).

Proof. We shall prove this by induction on n. The case n = 0 is true because

`(〈a0, a1〉) = δ(a0, a1).

Suppose that the result is true for all n < k, for some integer k > 0. Given a

chain 〈a0, . . . , ak+1〉, suppose that r0, . . . , rk ∈ Q>0 and δ(ai, ai+1) < ri for all i ≤ k.

We have to show that δ(a0, ak+1) <∑k

i=0 ri.

Let s =∑k

i=0 ri. It follows from the decidability of < on the rationals that there

is a greatest j ∈ 0, . . . , k such that∑j−1

i=0 ri <12s. For this value of j, we also have∑j

i=0 ri ≥ 12s and hence

∑ki=j+1 ri ≤ 1

2s. So by the induction hypothesis, δ(a0, aj) < s

and δ(aj+1, ak+1) < s (except for the cases j = 0 and j = k, which are covered below).

Clearly δ(aj, aj+1) < s, so by Lemma 6.4.6

δ(a0, ak+1) ≤ 2 maxδ(a0, aj), δ(aj, aj+1), δ(aj+1, ak+1) < 2s

For the case j = 0, we have δ(a0, a1) < s and δ(a1, ak+1) < s. It follows from the

definition of δ that δ(a0, a0) < δ(a0, a1) < s, so we may proceed as above. The case

j = k is similar.

Let V be the preuniformity on S induced by ν, that is, V = Uνε | ε ∈ Q>0.

Lemma 6.4.8 V is finer than U.


Proof. Given n ∈ N, we will show that Uν2−n−3 ≤ Un. Suppose that a ∈ S, Pos a and

ν(a, a) < 2−n−3. Since a is positive and Un+1 is a cover, there is a positive a′ / a such

that a′ / U for some U ∈ Un+1. Choose U ′ ∈ Un such that StUn+1(a′) / U ′. We will

prove that Bν2−n−3(a) / U ′.

Suppose that b ∈ Bν2−n−3(a). Since Un+2 is a cover, and using Positivity, we can

obtain

b / b′ ≤ b | Pos b′ & (∃V ∈ Un+2)[b′ / V ] (∗)

Given b′ in this set, since ν(a, b) < 2−n−3 there is a chain 〈c0, . . . , cm+1〉 from a′ to

b′ of length less than 2−n−3, and hence δ(c0, cm+1) < 2−n−2 by Lemma 6.4.7. So

c0, cm+1 / W for some W ∈ Un+2. W ⊆ StUn+2(a′), so (making use of Lemma 6.2.2)

b′ / StUn+2(cm+1) ⊆ StUn+2(StUn+2(a′)) / StUn+1(a

′) / U ′

for each b′ in the RHS of (∗).

Lemma 6.4.9 U is finer than V.

Proof. Given ε ∈ Q>0, choose n ∈ N such that 2−n < ε. We will show that Un ≤ Uνε .

Suppose that U ∈ Un. By the assumption that Un = pUn, there is an a ∈ U such

that Pos a. For each b ∈ U we have δ(a, b) ≤ 2−n < ε. It follows that ν(a, b) < ε for

all positive b ∈ U , so U / Bνε (a) ∈ Uν

ε .

This completes the proof of Theorem 6.4.4. It is easy to prove that, if we allow

the axiom of Dependent Choices, given any countable uniformity U we can choose a

sequence (U ′n)n∈N of covers satisfying the conditions of the theorem such that U′ =

U ′n |n ∈ N is equivalent to U. The metrisation result from [67] can then be proved.

6.4.3 Gauges

In this subsection, we shall give a definition of a gauge on a formal topology using

basic diameters, describe the (covering) uniformity it induces, and give sufficient

conditions under which we can find a gauge for a given uniformity.


Gauges and their induced (pre)uniformities

Given a family (Ui)i∈I of preuniformities, let U = Ui1 ∧ · · · ∧ Uin | i1, . . . , in ∈ I,

where U ∧ V = U ↓V |U ∈ U & V ∈ V. It can be shown that U is a preuniformity,

and that it is the coarsest preuniformity which is finer than each Ui. We shall call U

the coarsest refinement of (Ui)i∈I .

Even if Ui is proper for each i ∈ I, there seems to be no obvious way of proving

that the coarsest refinement of (Ui)i∈I is proper. It can be shown that if U′ =

p(Ui1 ∧ · · · ∧ Uin) | i1, . . . , in ∈ I then U′ is the coarsest proper uniformity which is

finer than each Ui; we call U′ the coarsest proper refinement of the family (Ui)i∈I .

Definition 6.4.10 A gauge on an open formal topology S is a set-indexed family

(νi)i∈I of basic diameters on S.

Given a gauge (νi)i∈I , let Ui be the preuniformity induced by νi for each i ∈ I.

The preuniformity induced by the gauge (νi)i∈I is the coarsest proper refinement of

the family (Ui)i∈I .

Classically the preuniformity induced by a gauge was defined to be the coarsest

refinement of the family of preuniformities induced by the individual diameters, and

this was used as the definition in an earlier draft of the material in this section.

However we found that, using that definition, uniformly continuous maps between

uniform formal topologies did not appear to correspond to the most natural definition

of uniform continuity for maps between gauge spaces. It therefore seems that the

coarsest proper refinement should be used instead. Classically all the preuniformities

involved are proper, so the two definitions coincide.

Proposition 6.4.11 Up to equivalence, the preuniformity induced by a gauge (νi)i∈I

is given by

U = Uνi1,...,νin

ε | i1, . . . , in ∈ I

where Uν1,...,νnε = Bν1,...,νn

ε (a) | a ∈ Pos & (∀k ≤ n)[νk(a, a) < ε]

and Bν1,...,νnε (a) = b ∈ S | (∀k ≤ n)[νk(a, b) < ε]


Proof. For convenience, given a list i = 〈i1, . . . , in〉 of elements of I, we shall write

U iε for Uνi1

,...,νinε , and B i

ε(a) for Bνi1

,...,νinε .

First we have to check that U is a proper preuniformity.

U1: Given i1, . . . , in, j1, . . . , jm ∈ I and ε, δ ∈ Q>0, let k = 〈i1, . . . , in, j1, . . . , jm〉

and η = minε, δ. Then Bkη (a) ⊆ B i

ε(a) ∩ Bjδ(a) for all a ∈ S, and hence

U kη ≤ U i

ε,Ujδ .

U2: Given ε ∈ Q>0 and i1, . . . , in ∈ I, we will show that U iδ <

∗ U iε, where δ = 1

3ε.

Suppose that a ∈ Pos, νik(a, a) < δ for all k ≤ n and b ∈ StU iδ(B i

δ(a)). There is

a c ∈ Pos such that b ∈ B iδ(c) and Pos(B i

δ(c) ↓B iδ(a)). Choose a d ∈ Pos such

that d ∈ B iδ(c)↓B i

δ(a).

Given k ≤ n, νik(a, b) ≤ νik(a, d) + νik(d, c) + νik(c, b) < δ + δ + δ = ε. So

b ∈ B iε(a).

U4: Clearly U = pU for all U ∈ U.

This completes the proof that U is a proper preuniformity. Clearly for each i ∈ I,

U is finer than the preuniformity induced by νi. Now suppose that V is another

proper preuniformity which is finer than each induced preuniformity. We have to

show that V is finer than U.

Given i1, . . . , in ∈ I and ε ∈ Q>0, let δ = 12ε. There exist V1, . . . ,Vn ∈ V such

that Vk ≤ Uνikδ for each k ≤ n. By U1 there is a V ∈ V such that V ≤ Vk for each k,

and by U4 there is a V ′ ∈ V such that V ′ ≤ pV . We will show that pV ≤ U iε, so that

V ′ ≤ U iε.

Given V ∈ pV , since V refines each Uνikδ there exist a1, . . . , an ∈ Pos such that

V / Bνi1δ (a1) ↓ · · · ↓Bνin

δ (an). By Monotonicity there is a b ∈ Bνi1δ (a1) ↓ · · · ↓Bνin

δ (an)

such that Pos b. If c ∈ Bνi1δ (a1)↓· · ·↓Bνin

δ (an) then for each k ≤ n we have

νik(b, c) ≤ νik(b, ak) + νik(ak, c) < δ + δ = ε

so c ∈ B iε(b). Thus V / B i

ε(b) ∈ U iε.


Lemma 6.4.12 Given a ∈ Pos, b ∈ S, ε ∈ Q>0 and i1, . . . , in ∈ I,

(i) if a, b ∈ U for some U ∈ U iε, then νik(a, b) < 2ε for all k ≤ n.

(ii) if νik(a, b) < ε for all k ≤ n then a, b ∈ U for some U ∈ U iε.

Proof. (i) U = B iε(c) for some c ∈ Pos, so νik(a, b) < 2ε for all k ≤ n by ν3.

(ii) a ∈ Pos and νik(a, a), νik(a, b) < ε for all k ≤ n, so a, b ∈ B iε(a) ∈ U i

ε.

Corollary 6.4.13 Given U ∈ Pow(S), ε ∈ Q>0 and i1, . . . , in ∈ I,

StU iε(U) ⊆ b ∈ S | (∃a ∈↓U)[Pos a & (∀k ≤ n)[νik(a, b) < 2ε] ⊆ StU i

2ε(U)

Hence a ∈ uc(b) if and only if there exist ε ∈ Q>0 and i1, . . . , in ∈ I such that

(∀a′ ∈ Pos)(∀b′ ∈ S)[a′ ≤ a & (∀k ≤ n)[νik(a′, b′) < ε] → b′ / b]

A gauge on S is said to be compatible if the preuniformity it induces is a uniformity

on S.

Proposition 6.4.14 Let S and S ′ be open formal topologies with gauges (νi)i∈I and

(ν ′i)i∈I′ respectively. A formal topology map r : S → S ′ is uniform w.r.t. the induced

preuniformities if and only if for all j1, . . . , jm ∈ I ′ and ε ∈ Q>0 there exist i1, . . . , in ∈

I and δ ∈ Q>0 such that

(∀a, b ∈ Pos)(∀a′, b′ ∈ S ′)[ ara′ & brb′ & (∀k ≤ n)[νik(a, b) < δ]

→(∀l ≤ m)[ν ′jl(a′, b′) ≤ ν ′jl

(a′, a′) + ν ′jl(b′, b′) + ε] ] (∗)

Proof. Suppose that r : S → S ′ is uniform. Given j1, . . . , jm ∈ I ′ and ε ∈ Q>0,

choose i1, . . . , in ∈ I and δ ∈ Q>0 such that (∀U ∈ U iδ)(∃U ′ ∈ U ′jε/2)[U / r−U ′].

Let a, b ∈ Pos, a′, b′ ∈ S ′ and suppose that ara′, brb′ and (∀k ≤ n)[νik(a, b) < δ].

By Lemma 6.4.12 there is a U ∈ U iδ such that a, b ∈ U . Choose U ′ ∈ U ′jε/2 such that

U / r−U ′. Since a / r−(a′ ↓U ′) and Pos a, there is an a′1 ∈ (a′ ↓U ′) such that Pos a′1.


Similarly there is a b′1 ∈ (b′ ↓U ′) such that Pos b′1. U ′ = B jε/2(c

′) for some c′ ∈ Pos′,

so for each l ≤ m,

ν ′(a′, b′) ≤ ν ′jl(a′, a′1) + ν ′jl

(a′1, c′) + ν ′jl

(c′, b′1) + ν ′jl(b′1, b

′)

≤ ν ′jl(a′, a′) + 1

2ε+ 1

2ε+ ν ′jl

(b′, b′)

= ν ′jl(a′, a′) + ν ′jl

(b′, b′) + ε

so (∗) is satisfied.

Conversely suppose that r satisfies the RHS of the proposition. Given j1, . . . , jm ∈

I ′ and ε ∈ Q>0, choose i1, . . . , in ∈ I and δ ∈ Q>0 satisfying (∗) for j and 13ε. We

will show that (∀U ∈ U iδ)(∃U ′ ∈ U ′jε)[U / r−U ′].

Given U ∈ U iδ, U = B i

δ(c) for some c ∈ Pos. Since U ′jε/3 is a cover, we can choose

a′ ∈ Pos′ such that νjl(a′, a′) < 1

3ε for each l ≤ m and Pos(c ↓ r−B′j

ε/3(a′)). Choose

a ∈ Pos such that a ∈ c↓r−B′jε/3(a

′).

S / r−b′ ∈ S ′ | (∀l ≤ m)[νjl(b′, b′) < ε/3],

and it follows that U / r−U ′, where

U ′ = b′ ∈ S ′ | Pos(U ↓r−b′) & (∀l ≤ m)[νjl(b′, b′) < ε/3].

Given b′ ∈ U ′, choose b ∈ Pos such that b ∈ U ↓ r−b′. Since b ∈↓ U , we have

νik(a, b) < δ for each k ≤ n. So by (∗), ν ′jl(a′, b′) < ν ′jl

(a′, a′) + ν ′jl(b′, b′) + 1

3ε < ε.

Thus U ′ ⊆ B′jε(a

′) ∈ U ′jε.

6.4.4 Obtaining a gauge from a proper preuniformity

Theorem 6.4.15 Suppose that U = Ui | i ∈ I is a proper preuniformity, and that

there exists a function f : I → I such that Uf(i) <∗ Ui for each i ∈ I. Then U

is equivalent to the preuniformity induced by some family of finitary basic diameters

on S.

Classically this can be proved by taking the gauge consisting of all basic diameters

which induce a preuniformity coarser than U. In CZF this collection would not be


a set, but we can use Strong Collection to obtain a set containing enough of these

basic diameters to induce U.

The proof in [67] makes use of Dependent Choice; this can be avoided by assuming

the existence of the function f . In the presence of the Presentation Axiom (see [9] or

[2]), such a function always exists.

Lemma 6.4.16 If U satisfies the assumptions of Theorem 6.4.15 and U ∈ U, then

there is a sequence V0,V1,V2, . . . of covers of S such that V0 = pS, V1 = pU ,

(∀n ≥ 1)(∃U ′ ∈ U)[Vn = pU ′] and Vn+1 <∗ Vn for each n.

Proof. U = Ui for some i ∈ I. Let V0 = pS and Vn+1 = pUfn(i) for each n ∈ N.

For each j ∈ I, if U ∈ pUf(j) then PosU and StUf(j)(U) / V for some V ∈ Uj.

V ∈ pUj since PosU and U / V . Thus pUf(j) <∗ pUj.

Lemma 6.4.17 If U satisfies the assumptions of Theorem 6.4.15, then there is a set

J of sequences of covers of S, such that

(i) for each sequence (Vn)n∈N ∈ J , V0 = pS, Vn = pVn and Vn+1 <∗ Vn for all n.

(ii) for every cover U ∈ U there is a sequence (Vn)n∈N ∈ J such that V1 = pU .

(iii) for every (Vn)n∈N ∈ J and n ≥ 1, there is a U ∈ U such that Vn = pU .

Proof. Let θ(U , F ) be the formula which asserts that

• F is a function N → Cov(S)

• F (0) = pS

• F (1) = pU

• F (n+ 1) <∗ F (n) for all n ∈ N

• For all n ≥ 1 there is a U ′ ∈ U such that F (n) = pU ′.


By Lemma 6.4.16, (∀U ∈ U)(∃F )θ(U , F ). Thus by Strong Collection there is a set J

such that (∀U ∈ U)(∃F ∈ J)θ(U , F ) and (∀F ∈ J)(∃U ∈ U)θ(U , F ), and it follows

that (i), (ii) and (iii) hold for J .

Proof of Theorem 6.4.15. Let J be the set of sequences whose existence was

proved in Lemma 6.4.17. For each F ∈ J , VF = F (0), F (1), F (2), . . . is a proper

preuniformity on S. Let νF be the basic diameter which induces a preuniformity

equivalent to VF , constructed in Theorem 6.4.4. To prove that U is (equivalent to)

the preuniformity induced by the gauge (νF )F∈J , we will show that U is the coarsest

proper preuniformity which refines each VF .

Given F ∈ J and V ∈ VF , we have to show that U ≤ V for some U ∈ U.

V = F (n) for some n ∈ N; w.l.o.g. we may assume that n ≥ 1, since F (1) ≤ F (0) by

(i) of Lemma 6.4.17. By (iii), V = pU ′ for some U ′ ∈ U, and since U is proper there

must be a U ∈ U such that U ≤ pU ′ = V . Thus U is finer than each VF .

Now suppose that V is a preuniformity and that V refines each VF . If U ∈ U then

by (ii) there is an F ∈ J such that pU ∈ VF . Since V refines VF there is a V ∈ V

such that V ≤ pU ≤ U . Thus V is finer than U.

6.5 Uniform formal topology completions

In this section we will show how to generalize the results of Section 5.4 to uniform

formal topology completions, and fill in the missing details from the previous chapter.

Our construction is similar to that of Krız in [53], but we avoid classical logic, and

for the sake of predicativity some care must be taken when choosing a base for the

completion.

6.5.1 Adequate bases

As with metric formal topology completions, the completion of a uniform formal

topology will have the same base as the original formal topology, provided this base

is “adequate”.


Definition 6.5.1 A uniform formal topology S has an adequate base (for the uni-

form completion) if for all U ∈ U there is a V ∈ U such that

(∀V ∈ V)(∃U ∈ U)(∃a ∈ AU)[StV(V ) / a]

If necessary, we can construct an adequate base by adding AU to the base for

each U ∈⋃

U.

Proposition 6.5.2 If S is an open formal topology with a compatible basic diame-

ter ν, U is the uniformity induced by ν and S has an adequate base for the metric

completion, then S has an adequate base for the uniform completion.

Proof. Suppose that S has an adequate base for the metric completion. Given

ε ∈ Q>0, choose δ ∈ Q>0 such that

(∀b ∈ Pos)[ ν(b, b) < δ → (∃a ∈ S)[b ∈ mcδν(a) & ν(a, a) < ε] ]

Given V ∈ Uνδ/2, V = Bν

δ/2(b) for some b ∈ Pos with ν(b, b) < 12δ, and so there is a

a ∈ S such that b ∈ mcδν(a) and ν(a, a) < ε.

If b′ ∈ StUνδ/2

(V ) then ν(b, b′) < δ and hence b′ / a. Thus StUνδ/2

(V ) / a. Also

a ∈ Bνε (a) ∈ Uν

ε since ν(a, a) < ε.

6.5.2 Inductively generating the completion

Given a uniform formal topology S with an adequate base, we define its completion

to be S = IG0(〈S,≤′, C ′〉), where S is the underlying set of S, a ≤′ b iff a / b, and

C ′ : S → Pow(Pow(S)) is defined by

C ′(a) = b ∈ uc(a) | Pos b ∪ b ≤′ a | (∃U ∈ U)[b / U ] | U ∈ U

for each a ∈ S.

If the uniformity U is known to be inhabited, the same formal topology can be

inductively generated by taking C ′(a) to be

b ∈ uc(a) | (∃U ∈ U)[b / U ] & Pos b | U ∈ U


for each a ∈ S. It is easy to see from this that the metric and uniform completions

of a metric formal topology are identical.

The uniformity on U is given by U = U | U ∈ U, where U = AU |U ∈ U.

For the remainder of this section, we shall assume that U = AU for each U ∈ U and

U ∈ U, so that U = U.

Let /′ and A′ denote the covering relation and the saturation operator belonging

to S.

Proposition 6.5.3 A′U ⊆ AU for all U ∈ Pow(S). If a, b ∈ S then a / b iff

a /′ b.

Proof. The proof is similar to that of Proposition 5.4.2.

It follows that S is a subspace of S. The subspace inclusion is the formal topology

map i = 〈a, b〉 ∈ S × S | a ∈ Ab from S to S.

Proposition 6.5.4 S is a strongly dense subspace of S.

Proof. Almost identical to Proposition 5.4.3.

Proposition 6.5.5 U is a uniformity on S, and the sets uc(a) are equal for S and

S. U is proper if and only if U is proper.

Proof. Without loss of generality, assume that U = U. If a ∈ S and U ∈ Pow(S)

then a /′⋃

U by the definition of C ′. Thus U ∈ Cov(S) for all U ∈ U.

Since AU = A′U = U for each U ∈ U ∈ U, it is easy to see that U ≤ V in S if

and only if U ≤ V in S. Thus U satisfies U1 for S.

For all U, V ∈ Pos(S), Pos(U ↓ V ) if and only if Pos(U ↓′ V ), so for each U ∈ U

and V ∈ Pow(S), the set StU(V ) is the same in both formal topologies. Therefore

uc(a) is the same in S and S, and U satisfies U2 for S.

U3 follows from the definition of C ′.

Since refinement (≤) and Pos are the same in S and S, U satisfies U4 for S if and

only if U satisfies U4 for S.


Call a uniform formal topology complete if AU = A′U for all U ∈ Pow(S). Since

the definition of A′ only uses /′ with a singleton on the right-hand side, together

with uc and Pos which are the same in S as in S, the completion of a uniform formal

topology is always complete.

6.5.3 The universal property

The aim of this section is to prove the following universal property of uniform formal

topology completions, and in doing so fill in the missing details from the proof of

Theorem 5.4.5.

Theorem 6.5.6 Given a uniform formal topology map r : S1 → S2 between uniform

formal topologies with adequate bases, there is a unique formal topology map r : S1 →

S2 such that the square below commutes

S1⊂

i1 - S1

S2

r

?⊂

i2 - S2

r

?

where i1 and i2 are the subspace inclusions of S1 and S2 into their completions.

Furthermore the map r is uniform.

As with metric formal topologies, it follows that we can define a functor from

UFTop to the full subcategory of complete uniform formal topologies by S 7→ S on

objects and r 7→ r on morphisms.

If r exists, then it must be unique by Lemma 5.4.8 and the fact that i1 is strongly

dense and every uniform formal topology is 2-regular. To prove existence, given

r : S1 → S2 define

r = 〈a1, a2〉 ∈ S1 × S2 | a1 /′1 r

−(uc(a2))

Lemma 6.5.7 Given U2 ∈ U2, there is a U1 ∈ U1 such that

(∀U1 ∈ U1)(∃U2 ∈ U2)(∃a2 ∈ A2U2)[A1U1 ⊆ r−a2]


Proof. Given U2 ∈ U2, since S2 has an adequate base there is a V2 ∈ U2 such that

(∀V2 ∈ V2)(∃U2 ∈ U2)(∃a2 ∈ AU2)[StV2(V2) /2 a2]

and since r is uniform there is a U1 ∈ U1 such that (∀U1 ∈ U1)(∃V2 ∈ V2)[U1 /1 r−V2].

Suppose that U1 ∈ U1. Choose V2 ∈ V2 such that U1 /1 r−V2, and choose U2 ∈ U2

and a2 ∈ A2U2 such that StV2(V2)/2a2. Then U1/1r−V2/1r

−a2, so AU1 ⊆ r−a2

by FTM4.

Lemma 6.5.8 Given U2 ∈ U2, there is a U1 ∈ U1 such that

(∀U1 ∈ U1)(∃U2 ∈ U2)(∃a2 ∈ A2U2)[A1U1 ⊆ r−a2]

Proof. Given U2 ∈ U2, choose V2 ∈ U2 such that

(∀V2 ∈ V2)(∃U2 ∈ U2)(∃a2 ∈ A2U2)[StV2(V2) /2 a2]

By Lemma 6.5.7 there is a U1 ∈ U1 such that

(∀U1 ∈ U1)(∃V2 ∈ V2)(∃b2 ∈ A2V2)[AU1 ⊆ r−b2]

Given U1 ∈ U1, choose V2 ∈ V2 and b2 ∈ A2V2 such that A1U1 ⊆ r−b2, and then

choose U2 ∈ U2 and a2 ∈ A2U2 such that StV2(V2) /2 a2, so that b2 ∈ uc2(a2). Then

A1U1 ⊆ r−b2 ⊆ r−(uc(a2)) ⊆ r−a2.

Lemma 6.5.9 Given U2 ∈ U2 and a1 ∈ S1, there is a subset W2 ⊆ Pos2 such that

a1 /′1 r

−W2, (∀c2 ∈ W2)(∃U2 ∈ U2)[c2 /2 U2] and (∀a2 ∈ S2)[a1ra2 → W2 /2 StU2(a2)]

Proof. Given U2 ∈ U2 and a1 ∈ S1, let

W2 = c2 ∈ S2 | (∃U2 ∈ U2)[c2 ∈ A2U2 & Pos1(a1 ↓1 r−c2)]

Choose U1 ∈ U1 satisfying the conclusion of Lemma 6.5.8.

a1 /′1

(a1

y1

⋃AU1 |U1 ∈ U1

)∩ Pos1

/′1⋃A1U1 |U1 ∈ U1 & Pos1(a1 ↓1U1)


Suppose that U1 ∈ U1 and Pos1(a1 ↓1U1). There is a U2 ∈ U2 and a c2 ∈ A2U2 such

that A1U1 ⊆ r−c2. Since U1 ⊆ r−c2, we have Pos1(a1 ↓r−c2), so c2 ∈ W2. This

completes the proof that a1 /′1 r

−W2.

Now suppose that a2 ∈ S2, a1ra2 and c2 ∈ W2. Then c2 ∈ A2U2 for some U2 ∈ U2,

and Pos1(a1 ↓ r−c2), so Pos1(r−a2 ↓1 r

−c2) and hence Pos2(a2 ↓2 c2). Thus

c2 /2 StU2(a2).

Lemma 6.5.10 r is a uniform formal topology map from S1 to S2.

Proof. FTM1: To show that S1 /′1 r

−S2, assume without loss of generality that

PosS1, so that PosS2 and hence U2 is inhabited by U3 and Monotonicity.

Choose U2 ∈ U2. By Lemma 6.5.8 there is a U1 ∈ U1 such that⋃

U1∈U1A1U1 ⊆

r−⋃

U2∈U2A2U2 ⊆ r−S2. It follows from the inductive definition of A′ that

S1 /′1

⋃U1∈U1

A1U1, so S1 /′1 r

−S2 as required.

FTM2: Suppose that a1 ∈ r−(uc2(b2)) ∩ r−(uc2(c2)) for some b2, c2 ∈ S2. We have

to show that a1 /′ r−(b2 ↓′2 c2). There exist b′2 ∈ uc2(b2) and c′2 ∈ uc2(c2) such

that a1rb′2 and a1rc

′2, and there is a U2 ∈ U2 such that StU2(b

′2) /2 b2 and

StU2(c′2) /2 c2.

By Lemma 6.5.9 there is a subset W2 ⊆ S2 such that a1 /′1 r

−W2, W2 /2 StU2(b′2)

and W2 /2 StU2(c′2). Thus W2 ⊆ (b2 ↓′2 c3), so a1 /

′1 r

−(b1 ↓′2 c3).

FTM3: We have to show that (i) if a2 /2 b2 then r−a2 ⊆ r−b2, and (ii) if

X2 ∈ C ′2(a2) then r−a2 /′1 r−X2.

(i) If a2 /2 b2 then uc2(a2) ⊆ uc2(b2), so r−uc2(a2) ⊆ r−uc2(b2) and hence

r−a2 ⊆ r−b2.

(ii) First consider the case where X2 = b2 ∈ uc2(a2) | Pos2 b2. Let a1 ∈

r−uc2(a2). There exist a′2 ∈ S2 and U2 ∈ U2 such that a1ra′2 and StU2(a

′2)/2

a2. Choose V2 ∈ U2 such that V2 <∗ U2.

By Lemma 6.5.9 there is a subset W2 ⊆ Pos2 such that a1 /′1 r

−W2 and

W2 /2 StV2(a′2). For each c2 ∈ W2, StV2(c2) /2 StV2(StV2(a

′2)) /2 StU2(a

′2) /2


a2. So W2 ⊆ X2 and hence a1 /′1 r

−X2.

The other case to consider is when X2 = b2 ≤′2 a2 | (∃U2 ∈ U2)[b2 ∈

A2U2] for some U2 ∈ U2. Let a1 ∈ r−uc2(a2). There exist a′2 ∈ S2 and

U2 ∈ U2 such that a1ra′2 and StU2(a

′2) /2 a2. By Lemma 6.5.9 there is a

subset W2 ⊆ S2 such that a1 /′1 r

−W2, W2 /2 a2 and (∀c2 ∈ W2)(∃U2 ∈

U2)[c2 /2 U2]. W2 ⊆ X2, so a1 /′1 r

−X2.

Uniform: This follows directly from Lemma 6.5.8.

Lemma 6.5.11 The square in Theorem 6.5.6 commutes.

Proof. Given a2 ∈ S2, we have to show that i−1 r−a2 =S1 r

−i−2 a2.

But r−a2 =S1 r−(uc2(a2)), so

i−1 r−a2 =S1 r

−a2 =A′r−(uc2(a2)) =S1 r−(uc2(a2))

=S1 r−a2 =S1 r

−A2a2 = r−i−2 a2

6.5.4 The points of uniform formal topology completions

As with metric spaces, we can obtain a completion operation on uniform spaces by

defining X = PtU(ΩUX ).

Cauchy points and complete uniform spaces

Let X be a standard uniform space, with uniformity U, such that ΩUX has an

adequate base for the uniform completion. A Cauchy point of X is a formal point of

ΩUX . It can be shown that α ∈ Pow(S) is a Cauchy point if and only if

(i) (∀U ∈ U)(∃a ∈ α)(∃U ∈ U)[ext a ⊆ extU ] and U is inhabited

(ii) (∀a, b ∈ α)(∃c ∈ α)[c ∈ uc(a) ∩ uc(b)]

(iii) (∀a ∈ α)(∀b ∈ S)[ext a ⊆ ext b→ b ∈ α]


(iv) (∀a ∈ α)[∃x ∈ ext a]

where uc(a) = b ∈ S | (∃U ∈ U)[StU(ext b) ⊆ ext a].

The assumption that U is inhabited ensures that α satisfies FP1. Conversely if α

is a formal point of ΩUX then there is a positive a ∈ α, so U is inhabited by U3 and

Monotonicity.

Definition 6.5.12 A Cauchy point α ⊆ S converges if α = αx for some x ∈ X. X

is complete if every Cauchy point converges.

Given any point x ∈ X, αx is a formal point of ΩUX , which is a subspace of ΩUX ,

so αx is a formal point of ΩUX , that is, a Cauchy point of X . Thus X is complete if

and only if the map x 7→ αx is a bijection between X and X.

Since every sober space is T0, X must be separated, and hence T3. Thus if α and

β are Cauchy points of X and α ⊆ β, then α = β.

If X is an Ru-metric space, then the Cauchy points of X (as an Ru-metric space)

are precisely the Cauchy points of the uniform space arising from X .

Cauchy points versus Cauchy filters

Completeness in uniform spaces (with entourages) is usually defined in terms of

Cauchy filters ; for more details see [44]. Under the correspondence between entourage

uniformities and covering uniformities, we obtain the following definition:

Definition 6.5.13 Given a small uniform space X , a Cauchy filter on X is a sub-

class F ⊆ Pow(X) such that F is a filter on Pow(X), each element of F is inhabited

and

(∀U ∈ U)(∃U ∈ U)[extU ∈ F ]

A Cauchy filter converges if there is a point x ∈ X such that (∀a ∈ αx)[ext a ∈ F ].

In [44], a uniform space was defined to be complete if every Cauchy filter converges,

and the points of the completion were the Cauchy filters which are minimal when

ordered by inclusion.


Proposition 6.5.14 If X is a small uniform space, ΩUX has an adequate base and

F is a Cauchy filter on X , then

αF = a ∈ S | (∃a′ ∈ uc(a))[ext a′ ∈ F ]

is a Cauchy point of X .

Proof. (i) Given U ∈ U, choose V ,W ∈ U such that (∀V ∈ V)(∃U ∈ U)(∃a ∈

AU)[StV(extV ) ⊆ ext a] and (∀W ∈ W)(∃V ∈ V)(∃b ∈ AV )[StW(extW ) ⊆

ext b]. There is a W ∈ W with extW ∈ F , so there exist V ∈ V , b ∈ AV ,

U ∈ U and a ∈ AU such that StV(extV ) ⊆ ext a and StW(extW ) ⊆ ext b. Thus

b ∈ uc(a) and ext b ∈ F .

(ii) Let a, b ∈ αF . There exist a′ ∈ uc(a) and b′ ∈ uc(b) such that ext a′, ext b′ ∈ F .

Choose U ∈ U such that StU(ext a′) ⊆ ext a and StU(ext b′) ⊆ ext b, and then

choose V ∈ U such that V <∗ U . By (i) there exist V ∈ V and c ∈ AV

such that c ∈ αF . Since F is a filter and each element of F is inhabited,

ext a ∩ ext c and ext b ∩ ext c are both inhabited. Hence ext c ⊆ StV(ext a′) and

so StV(ext c) ⊆ StU(ext a′) ⊆ ext a, so that c ∈ uc(a). Similarly c ∈ uc(b).

(iii) and (iv) are trivial.

It remains to prove that αF is a set. By (i), we have

(∀U ∈ U)(∃a)[a ∈ αF & (∃U ∈ U)[ext a ⊆ extU ]]

so by Strong Collection there is a set β such that

(∀U ∈ U)(∃a ∈ β)[a ∈ αF & (∃U ∈ U)[ext a ⊆ extU ]]

and (∀a ∈ β)(∃U ∈ U)[a ∈ αF & (∃U ∈ U)[ext a ⊆ extU ]]

so β ⊆ αF and β satisfies (i) of the definition of a Cauchy point. If a ∈ αF , then

there is a b ∈ αF such that StU(ext b) ⊆ ext a for some U ∈ U, and there is a c ∈ β

satisfying ext c ⊆ extU for some U ∈ U . Since c ∈ αF , ext b ∩ ext c is inhabited and

so ext c ⊆ StU(ext b) ⊆ ext a. Thus

αF = a ∈ S | (∃b ∈ β)[ext b ⊆ ext a]


so αF is a set by restricted separation.

Proposition 6.5.15 If α is a Cauchy point of a small uniform space X , then

Fα = F ∈ Pow(S) | (∃a ∈ α)[ext a ⊆ F ]

is a minimal Cauchy filter on X .

Proof. It follows from (ii) that F is a filter, and by (iv) each F ∈ F is inhabited.

Given U ∈ U, there is an a ∈ α such that ext a ⊆ extU for some U ∈ U , and so

extU ∈ Fα. Thus Fα is a Cauchy filter.

For minimality, suppose that F is a Cauchy filter and F ⊆ Fα. Let F ∈ Fα,

so ext a ⊆ F for some a ∈ α. By (ii) there is a b ∈ α and a U ∈ U such that

StU(ext b) ⊆ ext a. extU ∈ F for some U ∈ U . From ext b, extU ∈ Fα it follows that

ext b G extU and hence extU ⊆ ext a ⊆ F . Thus F ∈ F .

If α is a Cauchy point, then α ⊆ αFα by (ii), and αFα ⊆ α by (iii), so α = αFα .

Conversely, given a Cauchy filter F , we have FαF ⊆ F . It follows that if F is a

minimal Cauchy filter then FαF = F .

Proposition 6.5.16 Under the conditions of Proposition 6.5.14, F converges to a

point x ∈ X if and only if αF converges to x. Hence X is complete if and only if

every Cauchy filter converges.

Proof. If αF converges to x then clearly FαF converges to x, so F converges to x

since FαF ⊆ F . Conversely suppose that F converges to x. Given a ∈ αx, there

is a b ∈ αx with b ∈ uc(a). ext b ∈ F , so a ∈ αF . Thus αx ⊆ αF , and therefore

αx = αF .

Completeness of PtUS

Let S be a complete uniform formal topology. We may assume that S = S, U = U

and S has an adequate base. Let X ′ = PtUS; we will show that X ′ is a complete


uniform space. To avoid confusion we will use St′U(Y ), St′U(U), uc′(a) etc. when

working in X ′ or ΩUX ′, and use StU(U), uc(a) etc. for the corresponding notions

in S.

Lemma 6.5.17 Given a Cauchy point α′ of X ′, let

α = a ∈ S | (∃a′ ∈ uc(a))[a′ ∈ α′]

Then α is a formal point of S.

Proof. It suffices to prove the following:

(i) (∀U ∈ U)(∃U ∈ U)[AU G α] (and U is inhabited)

(ii) (∀a, b ∈ α)(∃c ∈ α)[c ∈ uc(a) ∩ uc(b)]

(iii) (∀a ∈ α)(∀b ∈ S)[a / b → b ∈ α]

(iv) α ⊆ Pos

(i) Given U ∈ U, choose V ,W ∈ U such that

(∀V ∈ V)(∃U ∈ U)(∃a ∈ AU)[V / a]

and (∀W ∈ W)(∃V ∈ V)(∃b ∈ AV )[W / b]

There is a c′ ∈ α′ such that ext′ c′ ⊆ ext′W ′ for some W ′ ∈ W . Pos′ c′, so there

is a formal point β of S such that c′ ∈ β.

Choose c ∈ β such that c ∈ AW for some W ∈ W. There exist U ∈ U , V ∈ V,

a ∈ AU and b ∈ AV satisfying StW(W )/b and StV(V )/a, so in particular

b ∈ uc(a). To prove that b ∈ α′, we will show that ext′ c′ ⊆ ext′ b.

Since β ∈ ext′ c ∩ ext′W ′, we have Pos(c↓W ′), so W ′ ⊆ StW(c) ⊆ Ab. Thus

ext′ c′ ⊆ ext′W ′ ⊆ ext′ b.

(ii) Suppose that a, b ∈ α, so there exist a′, b′ ∈ α′ and U ∈ U such that StU(a′)/a

and StU(b′)/b. Choose V ∈ U with V <∗ U . By (i) there is a c ∈ α such that

c ∈ AV for some V ∈ V .


It follows from (ii) and (iv) for α′ that ext′ a′ ∩ ext′ c is inhabited, so Pos(a′ ↓c).

Hence c/StV(a′) and StV(c)/StV(StV(a′))/StU(a′)/a, so c ∈ uc(a). Similarly

c ∈ uc(b).

(iii) and (iv) are trivial.

Theorem 6.5.18 If S is a complete uniform formal topology then PtU(S) is com-

plete.

Proof. Given a Cauchy point α′ of PtU(S), define α ∈ Pt(S) as in Lemma 6.5.17. α

and α′ are both Cauchy points of PtU(S), and α ⊆ α′, so α = α′. Thus every Cauchy

point of PtU(S) is an element of PtU(S).

Corollary 6.5.19 If X is a standard proper uniform space, then X = PtU(ΩU) is

complete.

6.6 Compactness and total boundedness

Definition 6.6.1 A uniform formal topology S is totally bounded if for all U ∈ U,

there exist U1, . . . , Un ∈ pU such that S / U1 ∪ · · · ∪ Un.

We will show that, under certain assumptions, the completion of a uniform formal

topology S is compact if and only if S is totally bounded. To see that this generalises

the corresponding result for metric formal topologies (Theorem 5.5.5) we have the

following:

Proposition 6.6.2 Let S be a metric formal topology with basic diameter ν, and let

U be the uniformity on S induced by ν. S is totally bounded w.r.t. ν if and only if

it is totally bounded w.r.t. U.

Proof. Suppose that S is totally bounded w.r.t. ν. Given ε ∈ Q>0, there exist

U1, . . . , Un ∈ Pow(S) such that PosUi and (∀a, b ∈ Ui)[ν(a, b) <12ε] for all i ≤ n,


and S /⋃n

i=1 Ui. For each i ≤ n there is an ai ∈ AUi with Pos ai and ν(ai, ai) <12ε.

Ui ⊆ Bνε (ai) for each i and hence S /

⋃ni=1B

νε (ai).

Conversely suppose that S is totally bounded w.r.t. U, and let ε ∈ Q>0. There

exist a1, . . . , an ∈ Pos such that ν(ai, ai) <12ε for each i and S /

⋃ni=1B

νε/2(ai). If

b, c ∈ Bνε/2(ai) then ν(b, c) ≤ ν(b, ai) + ν(ai, c) <

12ε + 1

2ε = ε. Thus S is totally

bounded w.r.t. ν.

Theorem 6.6.3 If S is a uniform formal topology with an adequate base, and U

is inhabited, then S is totally bounded if and only if its point-free completion S is

compact.

Proof. Let /′ be the covering relation from S.

Suppose that S is compact. If U ∈ U then S /′⋃AU |U ∈ U ∩ Pos, so there

exist a1, . . . , an ∈ Pos and U1, . . . , Un ∈ U such that ai ∈ AUi for all i ≤ n and

S /′ a1, . . . an. Thus S / a1, . . . , an / U1 ∪ · · · ∪ Un.

Conversely suppose that S is totally bounded. To prove that S is compact, we

will show that the set

θ = F ∈ FS | (∃F ′ ∈ FS)[S / F ′ & (∀a′ ∈ F ′)(∃a ∈ F )[a′ ∈ uc(a)]

satisfies the conditions 1–4 of Theorem 4.2.9 part (iii).

1. Suppose that F vL G and F ∈ θ. There is an F ′ ∈ FS such that S / F ′ and

(∀a′ ∈ F ′)(∃a ∈ F )[a′ ∈ uc(a)]. Clearly (∀a′ ∈ F ′)(∃b ∈ G)[a′ ∈ uc(b)], so

G ∈ θ.

2. Suppose that F ∈ FS, a ∪ F ∈ θ and X ∈ C(a). There is an F ′ ∈ FS such

that S / F ′ and (∀b′ ∈ F ′)(∃b ∈ a ∪ F )[b′ ∈ uc(b)].

We have to show that G ∪ F ∈ θ for some G ∈ FX. Either (i) X = b ∈

uc(a) | Pos b or (ii) X = b / a | (∃U ∈ U)[b / U ] for some U ∈ U.

(i) Choose U ∈ U such that (∀b′ ∈ F ′)(∃b ∈ a∪F )[StU(b′) / b], and choose

U ′ ∈ U with U ′ <∗ U . Since S has an adequate base, there exist V ,W ∈ U


satisfying

(∀V ∈ V)(∃U ∈ U ′)(∃a0 ∈ AU)[StV(V ) / a0]

and (∀W ∈ W)(∃V ∈ V)(∃b0 ∈ AV )[StW(W ) / b0] (∗)

Since S is totally bounded, there exist W1, . . . ,Wn ∈ pW with S/⋃n

i=1Wi.

For each i ≤ n, choose Vi ∈ V , bi ∈ AVi, Ui ∈ U ′ and ai ∈ AUi such that

StW(Wi) / bi and StV(Vi) / ai.

For each i ≤ n, Pos(bi ↓ c′i) for some c′i ∈ F ′, so Pos(ai ↓ c′i) and hence

StU ′(ai) / StU(c′i) / ci for some ci ∈ a ∪ F . Choosing c1, . . . , cn in this

way, we have ai ∈ uc(ci) for each i ∈ I. There is a decidable (finite) subset

I ⊆ 1, . . . , n such that for each i ≤ n, either

• ci = a and i ∈ I, or

• ci ∈ F and i /∈ I.

Let G = ai | i ∈ I and G′ = bi | i ∈ I. Then G ∈ FX, (∀b′ ∈ G′)(∃b ∈

G)[b′ ∈ uc(b)], and S / b1, . . . , bn / G ∪ F .

(ii) Choose U ′ ∈ U such that U ′ ≤ U and (∀b′ ∈ F ′)(∃b ∈ a∪F )[StU ′(b′)/b].

Since S has an adequate base, there exist V ,W ∈ U satisfying the two

equations (∗) above.

We can now argue exactly as in (i), noting that the set G = ai | i ∈ I

obtained does indeed satisfy (∀i ∈ I)(∃U ∈ U)[ai ∈ AU ], so that G ∈ FX.

3. By the assumption that U is inhabited, we may choose a cover U ∈ U. Choose

V ,W ∈ U satisfying (∗) above (with U ′ = U).

Since S is totally bounded, there exist W1, . . . ,Wn ∈ pW such that S / W1 ∪

· · · ∪ Wn. For each i ≤ n, choose Vi ∈ V , bi ∈ AVi, Ui ∈ U and ai ∈ AUi

satisfying StW(Wi) / bi and StV(Vi) / ai. Then bi ∈ uc(ai) for all i, and

S / b1, . . . , bn, so a1, . . . , an ∈ θ.

4. If F ∈ θ then S / F ′ for some F ′ ∈ FS satisfying (∀a′ ∈ F ′)(∃a ∈ F )[a′ ∈ uc(a)].

Clearly F ′ / F , so S / F .

Chapter 7

Balanced Formal Topologies and

the Basic Picture

7.1 Introduction to the Basic Picture

The Basic Picture Project is an ongoing project involving the preparation of a series

of papers by Giovanni Sambin and several others including Silvia Gebellato, Per

Martin-Lof and Venanzio Capretta. Three papers in the series [74, 76, 77] have

already appeared, and the author has received a draft of [78]; at least three more are

understood to be forthcoming. An introduction to the Basic Picture is also given by

Sambin and Gebellato in [75], and some of the concepts are discussed by Sambin in

[72]. Other relevant work includes [84] and [85] by Valentini, which deal with the

coinductive generation of the binary positivity predicate.

Some of the main novelties of the Basic Picture are:

• basic pairs as predicative generalised topological spaces,

• continuous relations between basic pairs as a generalisation of continuous func-

tions between spaces,

• the symmetry between the concrete and formal sides of a basic pair,

• the introduction of the interior operator J on the formal side of a basic pair,

213

CHAPTER 7. BALANCED FORMAL TOPOLOGIES 214

leading to the notion of a binary positivity predicate on a formal (basic) topol-

ogy.

In this chapter we aim to explore some aspects of the Basic Picture that have so

far received less attention. We shall mainly focus on functions between spaces rather

than relations between basic pairs, and on balanced formal topologies rather than the

more general formal basic topologies considered in [78] (essentially balanced formal

topologies which are not required to satisfy the axiom A3). The one exception to this

is Section 7.4, in which we give two counterexamples to an open problem involving

continuous relations between basic pairs.

Two topics which have been covered in earlier chapters are the adjunction be-

tween categories of topological spaces and balanced formal topologies (Subsection

3.2.3) which motivated the new notion of weak sobriety defined in Section 2.3, and

the equivalence between the category of balanced formal topologies and that of set-

generated FC-locales. The new categories of FC-frames and FC-locales may be of

some interest because they allow more direct comparisons to be made between the

properties of balanced formal topologies and ordinary frames and locales. We shall

pursue this further in Section 7.3, where we describe the limits and colimits of the

category of FC-frames in the impredicative intuitionistic case, and look at the notion

of “sublocales” of FC-locales.

Changing the binary positivity predicate

Assuming classical logic, if S is a formal topology with saturation operator A, a

binary positivity predicate can be obtained by taking JU = (AUC)C for each U ∈

Pow(S). The basic formal topologies obtained in this way behave in the same way

as the corresponding formal topologies (in particular they have the same points and

morphisms), and furthermore the basic formal topology arising from a topological

space will always satisfy this duality condition. However, the axioms for balanced

formal topologies are not sufficient to prove that JU = (AUC)C , even classically.

Two examples of “non-standard” binary positivity predicates defined in [78] are the


improper positivity predicate, where JU = ∅ for all U , and the trivial positivity

predicates, described below.

Definition 7.1.1 A binary positivity predicate n is trivial if J V = J S whenever

J V is inhabited.

It can be shown that n is trivial if and only if there is a subset P ⊆ S such that

J V = a ∈ P |P ⊆ V

for all V ∈ Pow(S). P ∈ Pow(S) gives rise to a (trivial) binary positivity predicate

in this way if and only if P satisfies Monotonicity, that is, AU G P → U G P for all

U ∈ Pow(S).

One reason for adding a binary positivity predicate is to allow more information

about a space to be “remembered” constructively. This is illustrated by the difference

between the constructive strength of sobriety and weak sobriety: the balanced formal

topology arising from any T2 space contains enough information to reconstruct the

original space, since that space is weakly sober. However, because the axioms are

not sufficient ensure that n is uniquely determined by /, the effects of adding n go

beyond this, and can affect the properties of the space, even classically.

By FP3′, every formal point α of a balanced formal topology must be formal

closed, that is, α = Jα. If S and S ′ are two balanced formal topologies on the same

set S with AU = A′U and J ′U ⊆ JU for all U ∈ Pow(S), then every formal closed

set of S ′ is formal closed in S, and hence Pt(S ′) ⊆ Pt(S). Thus balanced formal

topologies with smaller positivity predicates may have fewer points. In particular, if n

is the improper positivity predicate, then S only has one formal closed subset, namely

∅, and so it cannot have any formal points, even if the underlying formal topology

(without n) does have points. This appears to contradict the claim made in [78]

that adding the improper binary positivity predicate to a formal topology allows the

theory of locales to be (impredicatively) included in the theory of balanced formal

topologies.

It is also claimed in [78] that the theory of open formal topologies can be included

in that of balanced formal topologies by adding the trivial binary positivity predicate


arising from Pos, that is, J V = a ∈ Pos | Pos ⊆ V for all V ∈ Pow(S); furthermore

[78] and [72] both assert that in this case the axiom FP3′ for a formal point α is

equivalent to α ⊆ Pos, which is always true if Pos satisfies Positivity and α satisfies

FP3. However, given a formal point α of the balanced formal topology, it follows

from FP1 and FP3′ that Jα is inhabited, so α = J S = Pos. Therefore there can

be at most one formal point when n is trivial, and in many cases FP2 will not be

satisfied by α = Pos and there will be no formal points.

It would be desirable to find some additional axioms to add to the theory of bal-

anced formal topologies which ensure that classically JU = (AUC)C . This problem is

discussed in the next section; however the only axioms considered so far have proved

to be too restrictive constructively.

7.2 Problems with finding “enough” axioms

In this section we shall examine the problem of adding new axioms to the definition

of a balanced formal topology (Definition 3.2.13) to try to ensure that classically

JU = (AUC)C for all U ∈ Pow(S). Ideally any new axiom(s) added should be

constructively satisfied by ΩBX for each balanced-standard ct-space X , and they

should also be satisfied by (co)inductively generated formal topologies as described

in [72] (that is, without additional axioms for n).

Two new axioms are considered individually. The first axiom has the effect that

n is uniquely determined by /, and the second causes / to be uniquely determined

by n. However, we shall see that each new axiom fails to constructively satisfy one

of the criteria in the preceding paragraph.

7.2.1 Determining n from /

One possible axiom which may be added is the following, which ensures that n is the

largest binary positivity predicate which is compatible with /.

JU =⋃V ∈ Pow(U) | (∀W ∈ Pow(S))[AW G V → W G V ] (∗)


for each U ∈ Pow(S). In general if we tried to define n in this way then JU may

not be a set for all U ∈ Pow(S); however the intention is that (∗) is an axiom that

must be satisfied by a binary positivity predicate n defined in some other way. In

the case where A is inductively generated, (∗) is true if and only if n is equal to

the coinductively generated binary positivity predicate with no extra axioms. This

is one approach suggested in the conclusion of [85], but Valentini notes that some

interesting mathematical structures may get lost as a result. We will show that it

cannot be constructively proved that for every balanced-standard ct-space X , ΩBX

satisfies (∗).

Proposition 7.2.1 If S is a balanced formal topology satisfying (∗) and α ⊆ S

satisfies FP3 of Definition 3.2.1, then α satisfies FP3 ′ of Definition 3.2.17.

Proof. Suppose that α satisfies FP3. If we take V = α, then by FP3

(∀W ∈ Pow(S))[AW G V → W G V ]

So α ⊆ Jα as required.

Corollary 7.2.2 If X is a weakly sober balanced-standard ct-space and ΩBX satisfies

(∗), then X is sober.

Proof. The ideal points of X are the subsets of S satisfying axioms FP1–FP3. The

strong ideal points are the subsets of S satisfying FP1, FP2 and FP3′. If ΩBX

satisfies (∗) then every ideal point is a strong ideal point, so X is sober if and only if

it is weakly sober.

Corollary 7.2.3 Let X be the ct-space of rational numbers with the Euclidean topol-

ogy. If ΩBX satisfies (∗) then the non-constructive principle LPO holds.

Proof. X is T2, so by Proposition 2.3.10 it is weakly sober. If (∗) is satisfied then X

is sober by Corollary 7.2.2, and this implies LPO by Theorem 2.3.5.


7.2.2 Determining / from n

In [72] it is remarked that, impredicatively, a covering relation / can be defined from

a binary positivity predicate n by setting

a / U ⇔ (∀V ∈ Pow(S))[an V → U n V ] (†)

for all a ∈ S and U ∈ Pow(S). We shall consider the effect of adding (†) to the

axioms for balanced formal topologies. Note that the implication from left to right

is equivalent to the Compatibility axiom.

Proposition 7.2.4 If X is a balanced-standard ct-space then ΩBX satisfies (†).

Proof. Suppose that a ∈ S, U ∈ Pow(S) and (∀V ∈ Pow(S))[an V → U n V ]. We

have to show that ext a ⊆ extU .

Given x ∈ ext a, let V = αx. Then x ∈ ext a ∩ restV , so a n V and hence

U n V . Therefore there is a point y ∈ extU such that αy ⊆ αx, and it follows that

x ∈ extU .

Although (†) is satisfied by spatial balanced formal topologies, we will see that in

CZF it cannot be shown to be satisfied by all inductively generated balanced formal

topologies without extra axioms for n.

Proposition 7.2.5 Let S be a balanced formal topology (not necessarily satisfying

(†)) with saturation operator A and reduction operator J . For each U ∈ Pow(S), let

a /′ U ⇔ (∀V ∈ Pow(S))[an V → U n V ]

and a /′′ U ⇔ (∀α ∈ PtB(S))[a ∈ α→ α G U ]

and let A′U = a ∈ S | a /′ U and A′′U = a ∈ S | a /′′ U for each U ∈ Pow(S)1.

For each U ∈ Pow(S), AU ⊆ A′U ⊆ A′′U .

Proof. AU ⊆ A′U is just the Compatibility axiom for S. A′U ⊆ A′′U is true

because α = Jα for all α ∈ PtB(S).

1We do not appear to be able to show that A′U and A′′U are sets, or that A′ satisfies A3.


Proposition 7.2.6 Given a complete metric formal topology S0 = S0, let C =

〈S,≤, C〉 be the covering system which inductively generates S0 (see Subsection 5.4.2),

and let S be the balanced formal topology coinductively generated by 〈S,≤, C,D〉,

where D(a) = ∅ for all a ∈ S. Note that S has the same covering relation / as S0,

and that by the remark following Proposition 4.5.6 Pt(S0) = PtB(S).

Assuming Dependent Choice, if we define A′ and A′′ as in Proposition 7.2.5, then

A′U = A′′U for all U ∈ Pow(S).

Proof. Suppose that a /′′ U and an V . We have to show that U n V .

Using Dependent Choice, and the fact that J V is C,D-compatible, we can con-

struct a sequence (an)n∈N of elements of J V such that a0 = a, ν(an, an) < 1n

and

an ∈ mcν(an−1) for all n ≥ 1.

Now let α = b ∈ S | (∃n ∈ N)[an ≤ b]. It is easy to check that α ∈ Pt(S0) =

PtB(S). Since a ∈ α and a /′′ U we have U G α, so an ∈↓U for some n ∈ N. Thus

U G J V .

Corollary 7.2.7 Assuming DC, if (†) is satisfied by every (co)inductively generated

balanced formal topology without extra axioms for n, then every metric formal topol-

ogy completion is spatial.

Proof. Given a complete metric formal topology S0, define S, /′ and /′′ as in Propo-

sition 7.2.6.

(†) is equivalent to the assertion that AU = A′U for all U ∈ Pow(S), so by

Proposition 7.2.6 if S satisfies (†) then AU = A′′U for all U . Since S0 and S have

the same formal points, it follows that S0 is spatial.

Corollary 7.2.8 Assuming DC, if (†) is satisfied by every (co)inductively generated

balanced formal topology without extra axioms for n, then the Fan Theorem holds.

Proof. Let X be the Cantor space 2N. If X is equipped with a suitable metric then

it is a complete metric space, so X ≡ X = Pt(ΩMX ).


X is totally bounded, so it follows from Proposition 5.5.4 and Theorem 5.5.5 that

the point-free completion ΩMX is compact. By Corollary 7.2.7 ΩMX is spatial, so X

is compact. Thus 2N is compact, so the Fan Theorem holds2.

It is shown in [70] that there is a realisability model of CZF in which DC and CT

both hold. Since Church’s Thesis is inconsistent with the Fan Theorem, it follows

that it cannot be shown in CZF that (†) is satisfied by every (co)inductively generated

formal topology without extra axioms for n.

7.3 The category FCFrm in IZF

The category FCLoc was introduced in Subsection 3.3.3, and is equivalent to the cat-

egory of balanced formal topologies when working in the impredicative intuitionistic

set theory IZF.

In this section we describe some of the properties of FCFrm, working in IZF

throughout. We begin by describing the forgetful functor H : FCFrm → Frm,

and constructing a left adjoint G and a right adjoint K to H. Assuming classical

logic, we can show that G has a left adjoint; it remains to be seen whether a similar

construction can be carried out without resorting to classical logic.

We also describe the limits and colimits in FCFrm, and consequently we can

give a characterisation of the regular monomorphisms in FCLoc. The next logical

step would be to investigate whether these limits and colimits can be constructed

predicatively in the case of (co)inductively generated balanced formal topologies, in

a similar way to Chapter 4; however we shall not attempt to do this here.

7.3.1 Functors between Frm and FCFrm

Given a frame A, we can define two different FC-frames with the same underlying

frame as follows. Let GA be the FC-frame in which every completely prime upper

2Alternatively, we can use the fact that the formal Cantor space ΩMX is spatial if and only ifthe Fan Theorem holds (see [39]).


section of A is formal closed, and let KA be the FC-frame in which the only formal

closed subset is the empty set. GA has the largest possible collection of formal closed

subsets, and KA has the smallest possible collection.

If f : A → B is a frame homomorphism, then f is also an FC-frame morphism

GA → GB because f−1(D) is a completely prime upper section of A whenever

D is a completely prime upper section of B. f is also an FC-frame morphism

KA → KB, because f−1(∅) = ∅. Thus G and K can be extended to give two

functors Frm → FCFrm.

In the other direction, let H : FCFrm → Frm be the forgetful functor which

takes an FC-frame to its underlying frame. If A is an FC-frame and B is a frame,

then a function f : A → B is an FC-frame morphism A → KB if and only if it is

a frame homomorphism HA → B, since f−1(∅) = ∅ is always formal closed in A.

Given a function g : B → A, g is an FC-frame morphism GB → A if and only if it is

a frame homomorphism B → HA. It follows from this that G a H and H a K.

If classical logic is assumed, we can find a left adjoint to the functor G : Frm →

FCFrm. Given an FC-frame A with a set C of formal closed subsets, let FA be the

frame presented by the following generators and relations.

FA = Frm〈A (qua frame) | a ≤ b whenever (∀C ∈ C)[a ∈ C → b ∈ C] 〉

Let ηA : A → GFA be the frame homomorphism which takes a ∈ A to the element

of FA corresponding to the generator a.

Proposition 7.3.1 (ZF) η is the unit of an adjunction F a G.

To prove that ηA is an FC-frame morphism, we have to show that η−1A (D) ∈ C for

each completely prime upper section of FA.

With classical logic, it can be shown that the completely prime upper sections of a

frame are precisely the complements of the principal lower sections. Thus if D ⊆ FA

is a completely prime upper section, then D = (↓ ηA(b))C for some b ∈ A. Suppose

that there is an a ∈ η−1A (D) such that (∀C ∈ C)[a ∈ C → C G η−1

A (DC)]. Then (∀C ∈

C)[a ∈ C →∨η−1

A (DC) ∈ C], so ηA(a) ≤ ηA(∨η−1

A (DC)) =∨ηA(a′) | ηA(a′) ≤


ηA(b) and hence ηA(a) ≤ ηA(b), contradicting ηA(a) ∈ D. So by contradiction,

(∀a ∈ η−1A (D))(∃C ∈ C)[a ∈ C & C ⊆ η−1

A (D)]

so η−1A (D) is a union of formal closed subsets of A, and is therefore formal closed.

This completes the proof that ηA is an FC-frame morphism A→ GFA.

Now suppose that A is an FC-frame with formal closed subsets C, B is a frame

and f : A → GB is an FC-frame morphism. Suppose also that a, b ∈ A and (∀C ∈

C)[a ∈ C → b ∈ C]. (↓ f(b))C is a completely prime upper section of B, so

C = f−1((↓f(b))C) ∈ C. b /∈ C, so a /∈ C, hence f(a) ≤ f(b).

f is a frame homomorphism A → B which respects the axioms defining FA, so

there is a unique frame homomorphism g : FA→ B such that f = g ηA (regarding

f and ηA as frame homomorphisms). Gg is an FC-frame morphism GFA → GB,

and f = Gg ηA in FCFrm.

Under this adjunction, the frame homomorphisms FA → 1 are in one-one cor-

respondence with the FC-frame homomorphisms A → G1, where A is an FC-frame

and 1 is the initial frame (or terminal locale). G1 is also the initial object of FCFrm

(or the terminal object of FCLoc), so the points of the FC-locale A correspond to

those of the locale FA.

7.3.2 Limits of FC-frames in IZF

Proposition 7.3.2 Given a diagram D : I → FCFrm, suppose that (pI : L →

DI)I∈I is a limit cone for D in the category of frames. If L is given the formal closed

subsets

C = C ∈ Pow(L) | (∀a ∈ C)(∃I ∈ I)(∃C ′ ∈ CI)[a ∈ p−1I (C ′) ⊆ C]

where CI is the set of formal closed subsets of DI for each I ∈ I, then the cone is a

limit cone in FCFrm.

Proof. If I ∈ I and C ′ ∈ CI , clearly p−1I (C ′) ∈ C. Thus each projection pI is an

FC-frame morphism.


Given another cone (fI : A → DI)I∈I, we have to show that there is a unique

FC-frame morphism f : A → L such that pI f = fI for all I ∈ I. By the universal

property of L in Frm, there is a unique frame homomorphism f : A → L such that

fI = pI f for each I ∈ I. It remains to prove that f is an FC-frame morphism.

Given C ∈ C, it follows from the definition of C that C is a union of some sets of

the form p−1I (C ′), for I ∈ I and C ′ ∈ CI . For each such set, f−1(p−1

I (C ′)) = f−1I (C ′),

which is formal closed in A since fI is an FC-frame morphism. Thus f−1(C) is formal

closed, since the formal closed subsets of A are closed under unions.

Corollary 7.3.3 If (pI : L → DI)I∈I is a limit cone for a diagram D in FCFrm,

then the underlying cone in Frm is a limit cone for the same diagram, and the formal

closed subsets of L are precisely those described in Proposition 7.3.2.

Proof. Let (p′I : L′ → DI)I∈I be the limit cone constructed in Proposition 7.3.2. By

the universal property of the two limits, there is a (unique) FC-frame isomorphism

f : L′ → L such that p′I = pI f for each I ∈ I. Since f is a frame isomorphism, it

follows that (pI)I∈I is a limit cone in Frm. A subset C ⊆ L is formal closed if and

only if f−1(C) is formal closed in L′; it follows that the formal closed subsets of L

are those described in Proposition 7.3.2.

7.3.3 Colimits of FC-frames in IZF

Proposition 7.3.4 Given a diagram D : I → FCFrm, suppose that (qI : DI →

L)I∈I is a colimit cocone for D in the category of frames. If L is given the formal

closed subsets

C = C ∈ Pow(D) | (∀I ∈ I)[q−1I (C) ∈ CI ]

where CI is the set of formal closed subsets of DI for each I ∈ I, then the cocone is

a colimit cocone in FCFrm.

Proof. It is clear from the definition of C that each injection qI is an FC-frame

morphism. Given another cocone (fI : DI → A)I∈I, by the universal property of the


colimit in Frm there is a unique frame homomorphism f : L→ A such that fI = fqI

for each I ∈ I. If C ′′ is a formal closed subset of A, then q−1I (f−1(C ′′)) = f−1

I (C ′′) is

formal closed in DI, so f−1(C ′′) ∈ C. Hence f is an FC-frame morphism.

Corollary 7.3.5 If (qI : DI → L)I∈I is a colimit cocone for a diagram D in FCFrm,

then the underlying cocone in Frm is a colimit cocone for the same diagram, and the

formal closed subsets of L are precisely those described in Proposition 7.3.4.

Proof. Similar to Corollary 7.3.3, using the universal property of the colimits.

7.3.4 Coequalisers and sublocales

The regular monomorphisms in Loc, that is, the regular epimorphisms in Frm, are

precisely the surjective frame homomorphisms (see, for example, [45]). These maps

are known as the sublocale inclusions. We will show that the regular epimorphisms

in the category of FC-frames are precisely the surjective frame homomorphisms

f : A → A′ such that A′ has the largest possible collection of formal closed sub-

sets which makes f an FC-frame morphism.

Proposition 7.3.6 If e : A→ A′ is a surjective frame homomorphism between FC-

frames, and each subset C ′ ⊆ A′ is formal closed in A′ if and only if e−1(C ′) is formal

closed in A, then e is a regular epic FC-frame homomorphism.

Proof. First take the following pullback in FCFrm:

Pf2 - A

A

f1

? e- A′

e

?

By Proposition 7.3.2, this is constructed by first taking the pullback in Frm, that is,

P = 〈a1, a2〉 ∈ A× A | e(a1) = e(a2)


with the product ordering, defining the projections by fi(〈a1, a2〉) = ai for i = 1, 2,

and then giving P the collection of formal closed subsets described in that proposition.

It can be shown that e is the coequaliser (in Frm) of f1 and f2.

By Proposition 7.3.4, if C ′ ⊆ A′ is formal closed in A′ iff e−1(C ′) is formal closed

in A, then e is an FC-frame morphism, and it is the coequaliser of f1 and f2 in

FCFrm.

We also have the following converse:

Proposition 7.3.7 If e : A → A′ is a regular epimorphism in FCFrm, then e is

surjective and each subset C ′ ⊆ A′ is formal closed in A′ if and only if e−1(C ′) is

formal closed in A.

Proof. Let e be the coequaliser of a pair of FC-frame morphisms B -- A. By

Corollary 7.3.5 e is a coequaliser in the category of frames (so it is surjective), and

A′ has the formal closed subsets described in Proposition 7.3.4.

An interesting consequence of this is that if we define a sublocale of an FC-locale

A to be a regular FC-locale monomorphism into A (regarding two sublocales as equal

if each factors through the other), then the sublocales of an FC-locale are in one-one

correspondence with the sublocales of the underlying locale, regardless of the formal

closed subsets.

7.4 Continuous relations between basic pairs

The notion of a continuous relation between (small) basic pairs was introduced and

analysed in some detail in [76]. In this section we shall give a brief introduction

to continuous relations between small basic pairs, and then consider an open prob-

lem from [76] concerning continuity, proving its equivalence with another question

involving the closure and interior operators before giving two counterexamples.


7.4.1 Continuous relations

Let X = 〈X, , S〉 and X ′ = 〈X ′, ′, S ′〉 be basic pairs, and let r ⊆ X × X ′ be

a binary relation. We say that r− is concrete open if r−Y ′ is an open subset of X

whenever Y ′ is an open subset of X ′ (or equivalently if r− ext a′ is open for all a′ ∈ S ′).

Similarly r∗ is said to be concrete closed if r∗Y ′ is a closed subset of X whenever Y ′

is a closed subset of X ′ (or equivalently if r∗ restV ′ is closed for all V ′ ∈ Pow(S ′).

If r− is concrete open, we also say that r is a continuous relation from X to X ′.

It can be shown that r is continuous if and only if there is a relation s ⊆ S ×S ′ such

that the square

X + - S

X ′

+r

? ′

+ - S ′

+s

?

commutes, that is, r− ext′ a′ = ext s−a′ for all a′ ∈ S ′.

This generalises the notion of continuous functions between basic pairs: given a

function f : X → X ′, we identify f with its graph 〈x, x′〉 ∈ X ×X ′ | f(x) = x′, so

that f−Y ′ = f ∗Y ′ = f−1(Y ′) for all Y ′ ∈ Pow(X ′).

It is shown in [76] that if r ⊆ X ×X ′ and r− is concrete open then r∗ is concrete

closed. The question of whether the converse is true was left as an open problem.

We provide two counterexamples below. The first shows that, in CZF, if r∗ concrete

closed implies that r− is concrete open then we may deduce the non-constructive

principle LPO. The basic pairs in this counterexample are not topological spaces,

so we give a second counterexample, this time for the restriction of the problem to

functions between topological spaces. The argument here is less straightforward, and

involves the use of realisability models of IZF, showing that the statement in question

is inconsistent with the principles of Constructive Recursive Mathematics.

We begin by showing that the problem is equivalent to a question involving the

closure and interior operators on a basic pair.


Proposition 7.4.1 Given a small basic pair X = 〈X, , S〉, the following are equiv-

alent:

(i) For any other small basic pair X ′ = 〈X, ′, S ′〉 with the same set of points, if

every closed subset in X ′ is closed in X then every open subset in X ′ is open

in X .

(ii) For any small basic pair X ′ = 〈X ′, ′, S ′〉, if f : X → X ′ and f ∗ is concrete

closed, then f− is concrete open.

(iii) For any small basic pair X ′ = 〈X ′, ′, S ′〉, if r ⊆ X × X ′ and r∗ is concrete

closed, then r− is concrete open.

Proof. (iii) ⇒ (ii) is trivial.

(ii) ⇒ (i): Suppose that (ii) holds for X , and let X = 〈X, ′, S ′〉 be another basic

pair with the same set of points such that every closed subset in X ′ is closed in

X . Let f : X → X be the identity map on X. Then f−Y = f ∗Y = Y for all

Y ∈ Pow(X), so f ∗ is concrete closed, and hence f− is concrete open by (ii),

that is, every open subset in X ′ is open in X .

(i) ⇒ (iii): Given a basic pair X ′ = 〈X ′, ′, S ′〉 and a relation r ⊆ X ×X ′, consider

the basic pair Xr = 〈X, r, S′〉, where r = ′ r, that is,

x r a′ ⇔ (∃x′ ∈ X ′)[xrx′ & x′ ′ a′ ]

The open sets in Xr are the subsets of X of the form r− ext′ U ′ (where U ′ ∈

Pow(S ′)), and the closed sets in Xr are those of the form r∗ rest′ U ′. Thus r−

is concrete open iff every open subset in Xr is open in X , and r∗ is concrete

closed iff every closed subset in Xr is closed in X .

7.4.2 A counterexample using basic pairs

One of the central observations in [74] was that if X = 〈X, , S〉 is a (small) basic

pair, then so is X− = 〈S, −, X〉, where s − a iff a s. The concrete closed sets


of X are precisely the formal open sets of X−, and the concrete open sets of X are

precisely the formal closed sets of X−, and vice versa.

Thus the assertion that part (i) of Proposition 7.4.1 holds for all small basic pairs

X is equivalent to the assertion that if X = 〈X, , S〉 and X ′ = 〈X ′, ′, S〉 are two

basic pairs which share the same set S on the right and every formal open subset in

X ′ is formal open in X , then every formal closed subset in X ′ is formal closed in X .

Proposition 7.4.2 If the equivalent conditions of Proposition 7.4.1 hold for all small

basic pairs X , then every weakly sober ct-space is sober.

Proof. Given a ct-space X , let X ′ = Sob(X ) = 〈X ′, ′, S〉 be the soberification of

X . By Proposition 2.3.3, AU = A′U for all U ∈ Pow(S), so X and X ′ have the

same formal open subsets. By the assumption that part (i) of 7.4.1 holds for X− and

X ′−, X and X ′ have the same formal closed subsets, and hence JU = J ′U for all

U ∈ Pow(S).

For each ideal point α of X and each a ∈ α, α ∈ ext′ a∩rest′ α and hence a ∈ J ′α.

Thus α = Jα for each ideal point of X , so every ideal point is a strong ideal point;

therefore if X is weakly sober then it must be sober.

In Section 2.3 we saw that if every weakly sober space is sober then LPO holds.

Thus it cannot be shown constructively that the equivalent conditions of Proposition

7.4.1 are true for all small basic pairs.

7.4.3 A topological counterexample

Here we give a second counterexample to the problem, this time using topological

spaces. Unfortunately this counterexample requires us to move to the impredicative

theory IZF, and the proof involves the use of realisability models of IZF. Nevertheless,

since every theorem of CZF is true in IZF, it still follows that Sambin’s open problem

cannot be proved for functions between ct-spaces in CZF.


Weakly open sets

For the purposes of this section, given a small basic pair 〈X, , S〉, we shall call a

subset Y ⊆ X weakly open if clZ G Y → Z G Y for each subset Z ⊆ X.

Proposition 7.4.3 (IZF) Given a small basic pair X = 〈X, , S〉, define X ′ =

〈X, ′, S ′〉, where S ′ is the set of weakly open subsets of X and x ′ Y iff x ∈ Y .

The closed subsets of X are precisely the closed subsets of X ′.

Proof. Clearly every open set in X is weakly open in X , and hence open in X ′. It

follows from the definition of closure that cl′ Z ⊆ clZ for each subset Z ⊆ X.

To show that clZ ⊆ cl′ Z, suppose that x ∈ clZ. If x ∈ ext′ Y for some Y ∈ S ′,

then Y is weakly open and Y G clZ, so Y G Z. Thus x ∈ cl′ Z.

A principle which is not provable in IZF

Let N+ = N∪ ∞, where ∞ is an extra point to be added to N such that the union

is disjoint. N+ can be ordered in the obvious way with

x ≤ y iff (x ≤ y in N) ∨ (y = ∞)

In this section only, we shall say that a subset Y ⊆ N+ is unbounded if (∀N ∈ N)(∃n ∈

Y )[n ≥ N ] (so for example the sets 2, 4, 6, 8, . . . and ∞ are both unbounded),

and cofinite if (∃N ∈ N)(∀n ∈ N+)[n ≥ N → n ∈ Y ]. In particular, every cofinite set

must contain the point ∞.

Classically if Y ⊆ N+ meets every unbounded set then Y is cofinite, because

otherwise the complement of Y would be unbounded.

Proposition 7.4.4 In IZF it cannot be proved that if Y ⊆ N+ meets every unbounded

set then Y is cofinite.

Proof. By [58], there is a realisability model for IZF in which Church’s Thesis

(CT) and Countable Choice (ACω) are true; we will show that the above principle is

inconsistent with IZF+CT+ACω.


We can construct a bijection between N and N+ by relabelling 0 as ∞, 1 as 0,

2 as 1 and so on. If n is the code for a recursive partial function N N, then

composing this partial function with the bijection between N and N+ we obtain a

partial function N N+, which we shall denote by fn. Conversely if f : N → N+ is

a total function then it follows from CT that there is an n ∈ N such that f = fn.

Call a total function f : N → N+ inflationary if f(n) ≥ n for all n ∈ N. Let

Y = fn(n) |n ∈ N and fn is an inflationary total function N → N+

If Z ⊆ N+ is unbounded, then by ACω and CT there is an n ∈ N such that fn is an

inflationary total function N → N+ and fn(m) |m ∈ N ⊆ Z, so fn(n) ∈ Y ∩ Z.

Thus Y meets every unbounded subset of N+.

Now suppose that Y is cofinite. There is an N ∈ N such that

(∀n ∈ N+)[n ≥ N → n ∈ Y ]

There are infinitely many values of n for which fn is not an inflationary total function,

so in particular there is an M ∈ N such that there are at least N + 1 distinct values

of n ≤M for which fn is not inflationary and total; call these values n1, . . . , nN+1.

Since k ∈ Y for all N ≤ k ≤ M , there exist M − N + 1 natural numbers

mN , . . . ,mM such that fmkis inflationary and total and fmk

(mk) = k for each

N ≤ k ≤ M . Since fmkis inflationary, we have mk ≤ k ≤ N for each k. Fur-

thermore mN , . . . ,mM , n1, . . . , nN+1 are all distinct, so we have found a set of M + 2

distinct natural numbers ≤M — a contradiction.

The topological counterexample

Let X = 〈X, , S〉, where

X = N+ ext 〈0, x〉 = x

S = 0, 1 × N ext 〈1, x〉 = y ∈ N+ | y ≤ x

X is a ct-space. Note that a subset Y ⊆ X is open if and only if

∞ ∈ Y → (Y is cofinite)


and that clY = Y ∪ ∞ |Y is unbounded for each Y ∈ Pow(S).

Lemma 7.4.5 A subset Y ⊆ X is weakly open if and only if

(∞ ∈ Y ) → (Y meets every unbounded subset of N+) (∗)

Proof. Suppose that Y ⊆ X is weakly open, ∞ ∈ Y and that Z ⊆ X is unbounded.

Then ∞ ∈ Y ∩ clZ, so Y G Z.

Now suppose that Y ⊆ X satisfies (∗), Z ⊆ X and x ∈ Y ∩ clZ. If x ∈ N then

x ∈ Y ∩Z, and if x = ∞ then either x ∈ Z or Z is unbounded; in each case Y G Z.

Proposition 7.4.6 (IZF) The weakly open sets form a topology on X.

Proof. The only non-trivial condition to check is that the intersection of two weakly

open sets is weakly open. Suppose that Y1, Y2 ⊆ X are weakly open, ∞ ∈ Y1 ∩ Y2

and Z ⊆ X is unbounded. For each N ∈ N, the set ZN = n ∈ Z |n ≥ N

is unbounded and so Y1 G ZN . Hence Y1 ∩ Z is unbounded, so Y1 ∩ Z G Y2 or

equivalently Y1 ∩ Y2 G Z.

Proposition 7.4.7 If every weakly open subset of X is open, then any subset of N+

which meets every unbounded set is cofinite.

Proof. Suppose that Y ⊆ N+ touches every unbounded set. Then Y is weakly open,

and since ∞ is unbounded ∞ ∈ Y . If Y is open then it must contain a basic open

neighbourhood of ∞, and so it must be cofinite.

Still working in IZF, let X ′ = 〈X, ′, S ′〉 be the basic pair described in Proposition

7.4.3, in which S ′ is the set of weakly open subsets of X and x ′ Y iff x ∈ Y . By

Proposition 7.4.6 X ′ is a ct-space, and by Proposition 7.4.3 X and X ′ have the same

closed subsets.

If Sambin’s open problem were true for functions between ct-spaces then by con-

sidering the identity function on X (regarded as a map X ′ → X ) we would be able

to deduce that every open subset of X ′ is open in X , i.e. every weakly open subset

of X is open. By Propositions 7.4.7 and 7.4.4 this is not provable in IZF.

Chapter 8

Further Work

Although this thesis makes contributions to a number of areas in predicative con-

structive topology, especially formal topology, there are several topics which we have

not fully explored. We conclude with some suggestions for future research based on

the findings of this thesis.

Firstly, although it is very unlikely that quasi-formal topologies will replace formal

topologies in the predicative approach to point-free topology, they may still have some

useful applications. It therefore seems worthwhile to try to generalise some more of

the standard results from formal topology to cover quasi-formal topologies. In partic-

ular, as we have shown that every quasi-formal topology can be inductively generated

using a class-covering system, one could investigate whether the constructions of lim-

its, colimits etc. for inductively generated formal topologies can be generalised to

quasi-formal topologies.

Even including the results of this thesis, very little work has been completed

on the intuitionistic theory of point-free uniform spaces, predicative or otherwise.

One question which still needs to be addressed is whether properness should form

part of the definition of a uniform formal topology. The only concrete examples of

uniformities that we have considered here are those induced by metrics, or by metric

(or basic) diameters. Another interesting class of uniformities are those arising from

topological groups in the point-sensitive case, or localic groups in the point-free case.

Having formulated a predicative point-free definition of a covering uniformity, the

232

CHAPTER 8. FURTHER WORK 233

time may be right to attempt to transfer the theory of localic groups to predicative

formal topology. There is also the potential to make use of quasi-formal topologies

here, as the definition of a localic group includes a morphism A× A→ A. Provided

the construction of binary products can be generalised to quasi-formal topologies,

a quasi-formal topology map S × S → S can be used, where S is a (quasi-)formal

topology.

We have left a number of questions unanswered concerning weakly closed sub-

spaces and the “fibrewise” separation axioms. In particular we have been unable

to show whether 1- and 2-regularity are constructively equivalent, so it is not clear

whether 2-regularity is a new separation property or just a more convenient form of

1-regularity. There is also a wide range of other fibrewise separation properties which

we have not had time to study predicatively (see [48]).

Finally, there is considerable scope for more research concerning binary positivity

predicates and the Basic Picture. While we have been able to illustrate the differences

between balanced formal topologies and ordinary formal topologies, we have still been

unable to find any way of applying balanced formal topologies to find new point-free

versions of classical results which would not otherwise be possible. It may be that

weakly closed subspaces can provide a better way of dealing with closed subsets in

formal topology — while binary positivity predicates can express that one closed

set covers another and that an open set touches a closed set, using weakly closed

subspaces we can also express, for example, that an open subspace covers a closed

subspace.

Appendix A

The axioms of CZF and its

extensions

A.1 Class notation

Throughout the thesis, and in defining the axioms of CZF in this appendix, it will

be convenient to employ class notation. This is described in some detail in [9], and

we summarise the main points below.

CZF is a first-order theory in a language with just one binary predicate symbol,

∈. A class A is specified by a formula φ and a free variable x; we denote this class

by

A = x |φ(x)

Membership of classes is defined by

a ∈ x |φ(x) ≡ φ(a)

If A and B are classes, we say that A is a subclass of B (written A ⊆ B) if and only

if (∀x)[x ∈ A→ x ∈ B]. A and B are equal (written A = B) if A ⊆ B and B ⊆ A.

If a is a variable (denoting a set), we can also regard a as the class a = x |x ∈ a.

Given a class A, we say that A is a set if (∃x)[A = x]. The Axiom of Extensionality

ensures that the class notation a = b (where a and b are sets regarded as classes) is

equivalent to the first order formula a = b.

234

APPENDIX A. THE AXIOMS OF CZF AND ITS EXTENSIONS 235

We may now introduce further notation to define classes, for example:

• ∅ = x | ⊥

• V = x | >

• a1, . . . , an = x | (x = a1) ∨ · · · ∨ (x = an)

• A ∪B = x |x ∈ A ∨ x ∈ B

•⋃A = x | (∃y ∈ A)[x ∈ y]

• Pow(A) = x |x ⊆ A

• x ∈ A |φ(x) = x |x ∈ A & φ(x)

• a+ = a ∪ a

• 〈a, b〉 = a, a, b

• A×B = a | (∃x ∈ A)(∃y ∈ B)[a = 〈x, y〉]

• N: the class of natural numbers (including 0), as defined in [7]

A.2 The axioms of CZF

The axioms of CZF are listed below, using class notation with the aim of making them

more readable. They can be found in their original form, without class notation, in

[1] and [9].

A formula is restricted if it only contains quantifiers that are restricted, i.e. of the

form ∀x ∈ a or ∃x ∈ a, where a is a set.

Pairing: (∀a)(∀b)(∃c)[c = a, b]

Union: (∀a)(∃b)[b =⋃a]

Infinity: (∃a)[(∃x)[x ∈ a] & (∀x ∈ a)[x+ ∈ a]]

Extensionality: (∀a)(∀b)[ (∀x)[x ∈ a↔ x ∈ b] → a = b ]


Set Induction: For any class A,

(∀a)[a ⊆ A→ a ∈ A] → A = V

Restricted Separation: For any restricted formula φ(x, . . .),

(∀ · · · )(∀a)(∃b)[b = x ∈ a |φ(x, . . .)]

Strong Collection: For any formula θ(x, y, . . .),

(∀ · · · )(∀a)[ (∀x ∈ a)(∃y)θ(x, y, . . .) → (∃b)B(x∈a, y∈b)θ(x, y, . . .) ]

Subset Collection:

(∀a)(∀b)(∃c)(∀u)[ (∀x ∈ a)(∃y ∈ b)θ(x, y, u) → (∃z ∈ c)B(x∈a, y∈z)θ(x, y, u) ]

where B(x∈a, y∈b)φ means

(∀x ∈ a)(∃y ∈ b)φ & (∀y ∈ b)(∃x ∈ a)φ

A.3 Choice principles and Relation Reflection

We list three choice principles which may be added to CZF (see [9]). These are valid

in the type-theoretic interpretation of CZF, but we prefer to avoid their use whenever

possible as they are not valid in topos logic.

Countable Choice (ACω): If (∀n ∈ N)(∃x)φ(n, x) then there is a function f with

domain N such that (∀n ∈ N)φ(f(n)).

Dependent Choices (DC): If ψ(x, y) is a formula, a is a set, b0 ∈ a and

(∀x ∈ a)(∃y ∈ a)ψ(x, y)

then there is a function f : N → a such that f(0) = b0 and

(∀n ∈ N)ψ(f(n), f(n+ 1))


Relativised Dependent Choices (RDC): Given a class A and a formula ψ(x, y),

if b0 ∈ A and

(∀x ∈ A)(∃y ∈ A)ψ(x, y)

then there is a function f : N → A such that f(0) = b0 and

(∀n ∈ N)ψ(f(n), f(n+ 1))

In some situations where we would normally use Dependent Choices, it is possible

to use one of the two “relation reflection” schemes mentioned below (also known as

“dependent collection” schemes in some of the literature). These were introduced in

[4, 5] and, along with their corresponding extension axioms below, they are needed

to coinductively define sets and classes in CZF, and to construct coequalisers of set-

presented Frames [6].

Relation Reflection Scheme (RRS): Given a class X and a subclass R ⊆ X×X

such that

(∀a ∈ X)(∃b ∈ X)[〈a, b〉 ∈ R]

for all a0 ∈ X there is a subset Y ⊆ X such that a0 ∈ Y and

(∀a ∈ Y )(∃b ∈ Y )[〈a, b〉 ∈ R]

RRS is also known as the Dependent Collection Scheme (DCS). There is also

a stronger scheme, defined in [5], known as the Strengthened Relation Reflection

Scheme (sRRS) or the Strengthened Dependent Collection Scheme (sDCS), which is

not needed in this thesis.

A.4 Extension axioms

The Regular Extension Axiom (REA)

Let [r : a > b] be the formula

r ⊆ a× b & (∀x ∈ a)(∃y ∈ b)[〈x, y〉 ∈ r]


and let [r : a > < b] be the formula

r ⊆ a× b & B(x∈a, y∈b)[〈x, y〉 ∈ r]

A set A is transitive if (∀a ∈ A)[a ⊆ A]. A is regular if it is transitive and (∀a ∈

A)SubColl(a,A,A), where SubColl(a, b, c) abbreviates the formula

(∀r)[ [r : a > b] → (∃b′ ∈ c)[r : a > < b′] ]

The Regular Extension Axiom (REA) asserts that every set is a subset of a regular

set.

The Weak Regular Extension Axiom (wREA)

A transitive set A is weakly regular if

(∀a ∈ A)(∀r)[ [r : a > A] → (∃b ∈ A)[r : a > b] ]

The Weak Regular Extension Axiom (wREA) asserts that every set is a subset of a

weakly regular set.

The Strong Regular Extension Axiom (sREA)

A set A is union-closed if (∀a ∈ A)[⋃a ∈ A]. A strongly regular set is a regular

set which is union-closed. The Strong Regular Extension Axiom (sREA) asserts that

every set is a subset of a strongly regular set. This axiom is sometimes called uREA

(for example, in [7]).

The RRP-Regular Extension Axiom (RRP-REA)

A set A has the Relation Reflection Property if for allX ∈ Pow(A) and R ∈ Pow(X×

X), if

(∀a ∈ X)(∃b ∈ X)[〈a, b〉 ∈ R]

then for each a0 ∈ X there is a set Y ∈ A such that a0 ∈ Y ⊆ X and

(∀a ∈ Y )(∃b ∈ Y )[〈a, b〉 ∈ R]


A is RRP-regular if it is regular and has the Relation Reflection Property. The

RRP-Regular Extension Axiom (RRP-REA) asserts that every set is a subset of an

RRP-regular set.

The Strong RRP-Regular Extension Axiom (sRRP-REA)

A set A is strongly RRP-Regular if it is strongly regular and has the Relation Reflec-

tion Property. The Strong RRP-Regular Extension Axiom (sRRP-REA) asserts that

every set is a subset of a strongly RRP-regular set.

RRP-REA is also known as DCSREA, and sRRP-REA has been called sDCSREA

and ∗-REA in the literature. There is also a strengthened Relation Reflection Prop-

erty, which gives rise to notions of sRRP-regularity and strong sRRP-regularity (de-

fined in [5], where they are called sDCS-regularity and strong sDCS-regularity).

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Index

C-closed, 89

R, R+, R], 34

Ti, T+i , T ]

i , 33, 34

♦ and , 30

Ru-metric, 130

Ru-pseudometric, 130

αx, 32

G, 30

ext and rest, 30

r, r−, r∗ and r−∗, 29

ACω, 236

adequate base

for the metric completion, 150

for the uniform completion, 200

balanced covering system, 124

local, 124

balanced formal topology, 44, 51

spatial, 55

standard, 44, 55

balanced formal topology map, 52

basic diameter, 135

compatible, 136

basic pair, 30

Cauchy filter, 206

Cauchy point

of a metric space, 154

of a uniform space, 205

Church’s Thesis, 11

class-covering system, 118

closure operator, 29

coinduction, 122, 124

compatibility, 52

compatible

basic diameter, 136

diameter, 133

complete

metric formal topology, 151

metric space, 155

uniform formal topology, 202

uniform space, 206

completely prime upper section, 60

continuous map

quasi-standard, 72

standard, 45

continuous relation, 226

contractive formal topology map, 139

contractive function, 132, 159

covering system, 89

class-, 118

248

INDEX 249

equivalence of, 89

local, 89

covering uniformity, 171

ct-space, 31

balanced-standard, 44, 54

open-standard, 44, 50

quasi-small, 32

quasi-standard, 71

small, 32

sober, 37

standard, 44, 45

weakly sober, 38

CZF, 235

DC, 236

diameter, 133

basic, 135

compatible, 133

metric, 134

entourage uniformity, 169

expansive formal topology map, 162

expansive function, 159

Fan Theorem, 11

FC-frame, 60

set-generated, 61

FC-locale, 60

formal closed set, 52, 215

formal point

of a balanced formal topology, 54

of a formal topology, 46

of a quasi-formal topology, 73

formal topology, 20, 41

compact, 87, 109

locally compact, 87

set-presentable, 98

spatial, 47

standard, 44, 46

formal topology map, 41, 49

contractive, 139

image, 77

strongly dense, 81

frame, 18

set-generated, 58

Galois adjunction, 45

gauge, 194

compatible, 196

geometric theory, 95

model, 96

ideal point, 36

inductive definition, 17

inductive definitions, 17

interior operator, 29

isometry, 159, 162

IZF, 15

Lindenbaum algebra, 95

locale, 19, 57

set-generated, 58, 59

set-open, 59

lower powerlocale, 109

INDEX 250

LPO, 16

Markov’s Prinicple, 11


complete, 151

Dedekind, 140

finitary, 140

standard, 144

totally bounded, 157

metric space, 128, 131

complete, 155

standard, 144


metric subspace, 165

monotonicity, 48, 97, 98

nucleus, 76

open formal topology, 44, 48

standard, 44, 50

open locale, 48

PEM, 16

positivity, 48

positivity predicate

binary, 51

improper, 215

trivial, 215

unary, 48

preuniformity, 182

induced by a basic diameter, 188

induced by a gauge, 194

product

binary, 102

set-indexed, 102

quasi-formal topology, 67

spatial, 74

standard, 69

quasi-formal topology map, 69

RDC, 237

REA, 17, 237

reduction operator

of a ct-space, 31

of a balanced formal topology, 52

regular formal topology, 83, 85

regular set, 238

REM, 16

restricted formula, 14, 235

RRP-REA, 238

saturation operator

of a ct-space, 31

of a formal topology, 41

set-presentation, 98

sober, 37

spatial, 47, 55, 74

sREA, 238

sRRP-REA, 239

star-refinement, 170, 182

strong ideal point, 38

sublocale, 76, 224

subspace of a formal topology, 76

INDEX 251

closed, 79

open, 79

strongly dense, 81

weakly closed, 80, 83, 106

totally bounded


metric space, 157


transition system, 111


complete, 202

proper, 183

standard, 185


uniform formal topology map, 183

uniform space, 176

+-separated, 180

]-separated, 180

complete, 206

separated, 179

standard, 184

uniformity, 176

covering, 171

entourage, 169

on a formal topology, 182

uniformly continuous, 170, 172, 177

upper powerlocale, 108

upper real, 128, 129

way below, 87

weak limit point, 34

weakly sober, 38

well-inside, 83–85

wREA, 17, 238

wREM, 16

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