1. Topological Nets and Filters Mark Hunnell

2. Outline 1.Motivations for Nets and Filters 2.Basic Definitions 3.Construction of Equivalence 4.Comparison and Applications

3. Motivations for Nets/Filters Bolzano-Weierstrass Theorem Characterization of Continuity Existence of sequences converging to limit points Countable and Bicompactness

4. 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

5. 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

6. 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 (x n ) is said to converge to x ∊ X if for every neighborhood U of x, ∃ n ∊ J such that n ≤ b implies x b ∊ U. Observation: If J is the set of natural numbers, these are the usual definitions of a sequence. Example: Sets by reverse inclusion

7. A Divergent Net X Directed Set J

8. X0X0 A Convergent Net X Directed Set J

9. Basic Filter Definitions Definition 1: A non-void collection ℬ of non-void subsets of a set X is a filter base if ∀ B 1, B 2 ∊ ℬ, B 1 ∩B 2 ⊇ B 3 ∊ ℬ. Definition 2: A filter is a non-void collection ℱ of subsets of a set X such that: 1. Every set containing a set in ℱ is in ℱ 2. Every finite intersection of sets in ℱ is in ℱ 3. ∅ ∉ ℱ

10. 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 ℬ converges to x 0 ∊ X if every neighborhood U of x 0 contains some set from ℬ.

11. A Divergent Filter B1B1 B2B2 B3B3 B4B4 A ∉ ℱ

12. A Convergent Filter X= B 0 B1B1 B2B2 B3B3 B4B4 X0X0

13. Construction of Associated Filters Proposition 1: Let {x α } α ∊ J be a net in a topological space X. Let E(α)= { x k : k ≥ α}. Then ℬ ({x α }) = {E(α) : α ∊ X} is a filter base associated with the net {x α }. Proof: Let E(α 1 ), E(α 2 ) ∊ ℬ ({x α }). Since J is a directed set, ∃ α 3 such that α 1 ≤ α 3 and α 2 ≤ α 3. E(α 3 ) ⊆ E(α 1 ) ∩ E(α 2 ), and therefore ℬ ({x α }) is a filter base.

14. Convergence of Associated Filters 1 Proposition 2: Let {x α } α ∊ J converge to x 0 ∊ X ({x α }→ x 0 ), then ℬ ({x α })→ x 0. Proof: Since {x α }→ x 0, then for every neighborhood U of x ∃ α such that α ≤ β implies that x β ∊ U. Then each E(α)= { x k : k ≥ α} contains only elements of U, so E(α) ⊆ U. Thus every neighborhood of x 0 contains an element of ℬ ({x α }), so ℬ ({x α })→ x 0.

15. Convergence of Associated Filters 2 X x1x1 x2x2 x3x3 x4x4 E1E1 E2E2 E3E3 E4E4

16. Construction of Associated Nets Proposition 3: Let ℬ = { E α } α ∊ A be a filter base on a topological space X. Order A = {α} with the relation α ≤ β if E α ⊇ E β. From each E α select an arbitrary x α ∊ E α. Then ж( ℬ ) = {x α } α ∊ A is a net associated with the filter base ℬ. Proof: A is directed since the definition of a filter base yields the existence of γ such that ∀ α, β ∊ A, E α ∩ E β ⊇ E γ. Therefore α ≤ γ and β ≤ γ, so A is directed. We now show that each x α ∊ 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( ℬ ) is a net on X.

17. Convergence of Associated Nets Proposition 4: If a filter base ℬ converges to x 0 ∊ X, then any net associated with ℬ converges to x 0. Proof: Let ж( ℬ ) be a net associated with ℬ. Then for every neighborhood U of x 0 ∃ E α ∊ ℬ such that E α ⊆ U. Then ∀ β ≥ α, E β ⊆ E α ⊆ U. Then ∀ x β ∊ E β, x β ∊ U. Therefore ж( ℬ )→ x 0.

18. Convergence of Associated Nets X x1x1 x2x2 x3x3 x4x4 B1B1 B2B2 B3B3 B4B4

19. Filter Advantages Associated filters are unique Structure (subsets of the power set) Formation of a completely distributive lattice –Compactifications, Ideal Points –Relevance to Logic

20. Net Advantages Direct Generalization of Sequences Carrying Information Moore-Smith Limits Riemann Integral (Partitions ordered by refinement)

21. Summary Filters Topological Arguments Set Theoretic Arguments Nets Analytical Arguments Information