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