Topological Nets and Filters

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