Topological Nets and Filters

Mark Hunnell

Outline

1. Motivations for Nets and Filters

2. Basic Definitions

3. Construction of Equivalence

4. Comparison and Applications

Motivations for Nets/Filters

•Bolzano-Weierstrass Theorem

•Characterization of Continuity

•Existence of sequences converging to limit points

•Countable and Bicompactness

First Countability Axiom

Definition 1: X satisfies the first countability axiom if every point has a countable basis of neighborhoods

Examples:

1. Metric Spaces

2. Finite Complement Topology

3. /ℝ ℕ

Basic Net Definitions

Definition 1: A partial order relation ≤ on a set A satisfies:

1.      a ≤ a ∀a A∊

2.      a ≤ b and b ≤ a implies a = b

3.      a ≤ b and b ≤ c implies a ≤ c

Definition 2: A directed set J is a set with partial order relation ≤ such that ∀a, b J, c J such that a ∊ ∃ ∊ ≤ c and b ≤ c

Basic Net Definitions

Definition 3: Let X be a topological space and J a directed set. A net is a function f: J → X

Definition 4: A net (xn) is said to converge to x X if for ∊every neighborhood U of x, n J such that n ∃ ∊ ≤ b implies xb ∊U.

Observation: If J is the set of natural numbers, these are the usual definitions of a sequence.

Example: Sets by reverse inclusion

A Divergent Net

X

Directed Set J

X0

A Convergent Net

X

Directed Set J

Basic Filter Definitions

Definition 1: A non-void collection of non-void subsets of a ℬset X is a filter base if ∀B1, B2 , ∊ ℬ B1∩B2 ⊇ B3 .∊ ℬDefinition 2: A filter is a non-void collection of subsets of a ℱset X such that:

1.      Every set containing a set in is in ℱ ℱ2.      Every finite intersection of sets in is in ℱℱ3.      ∅ ∉ ℱ

Basic Filter Definitions

Lemma 1: A filter is a filter base and any filter base becomes a filter with the addition of supersets.

Definition 3: A filter base converges to xℬ 0 X if every ∊neighborhood U of x0 contains some set from .ℬ

A Divergent Filter

B1 B2B3

B4

A ∉ ℱ

A Convergent Filter

X= B0

B1

B2

B3

B4X0

Construction of Associated Filters

Proposition 1: Let {xα}α J∊ be a net in a topological space X.

Let E(α)= { xk : k ≥ α}. Then ({xℬ α}) = {E(α) : α X} is a ∊filter base associated with the net {xα}.

Proof: Let E(α1), E(α2) ({x∊ ℬ α}). Since J is a directed set, ∃α3 such that α1 ≤ α3 and α2 ≤ α3 .

E(α3) E(α⊆ 1) ∩ E(α2) , and therefore ({xℬ α}) is a filter base.

Convergence of Associated Filters 1

Proposition 2: Let {xα}α J∊ converge to x0 X ({x∊ α}→ x0), then

({xℬ α})→ x0.

Proof: Since {xα}→ x0, then for every neighborhood U of x

α such that α ∃ ≤ β implies that xβ U. Then each ∊

E(α)= { xk : k ≥ α} contains only elements of U, so

E(α) U. Thus every neighborhood of x⊆ 0 contains an

element of ({xℬ α}), so ({xℬ α})→ x0.

Convergence of Associated Filters 2

X

x1 x2 x3 x4

E1 E2

E3

E4

Construction of Associated Nets

Proposition 3: Let = { Eℬ α} α A ∊ be a filter base on a topological

space X. Order A = {α} with the relation α ≤ βif Eα E⊇ β. From

each Eα select an arbitrary xα E∊ α. Then ж( ) = {xℬ α} α A ∊ is a net

associated with the filter base .ℬProof: A is directed since the definition of a filter base yields the existence of γ such that α,∀ β A, E∊ α ∩ Eβ E γ. ⊇Therefore α ≤ γ and β≤ γ, so A is directed. We now show that each xα X. Since each x∊ α was chosen from a subset of X, this

is clearly the case. Therefore the process constructs a function from a directed set into the space X, so ж( ) is a net on X.ℬ

Convergence of Associated Nets

Proposition 4: If a filter base converges to xℬ 0 X, then any ∊net associated with converges to xℬ 0.

Proof: Let ж( ) be a net associated with . Then for every ℬ ℬneighborhood U of x0 E∃ α such that E∊ ℬ α U. Then ⊆ ∀β≥ α,

Eβ E⊆ α U. Then x⊆ ∀ β E∊ β, xβ U. Therefore ж( )→ x∊ ℬ 0.

Convergence of Associated Nets

X

x1 x2 x3 x4

B1 B2

B3

B4

Filter Advantages

•Associated filters are unique

•Structure (subsets of the power set)

•Formation of a completely distributive lattice–Compactifications, Ideal Points–Relevance to Logic

Net Advantages

•Direct Generalization of Sequences

•Carrying Information

•Moore-Smith Limits

•Riemann Integral (Partitions ordered by refinement)

Summary

FiltersTopological Arguments

Set Theoretic Arguments

NetsAnalytical Arguments

Information