Topological Nets and Filters

1. Topological Nets and Filters Mark Hunnell

2. Outline Motivations for Nets and Filters Basic Definitions Construction of Equivalence 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 (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

7. A Divergent Net X Directed Set J

8. X0 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 ∀B1, B2 ∊ ℬ, B1∩B2 ⊇ B3 ∊ 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 x0 ∊ X if every neighborhood U of x0 contains some set from ℬ.

11. A Divergent Filter B1 B2 B3 B4 A ∉ ℱ

12. A Convergent Filter X= B0 B4 X0 B1 B2 B3

13. 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 ℬ({xα}) = {E(α) : α∊ X} is a filter base associated with the net {xα}. Proof: Let E(α1), E(α2) ∊ ℬ({xα}). Since J is a directed set, ∃ α3 such that α1 ≤α3 and α2 ≤α3 . E(α3) ⊆E(α1) ∩ E(α2) , and therefore ℬ({xα}) is a filter base.

14. Convergence of Associated Filters 1 Proposition 2: Let {xα}α∊J converge to x0 ∊ X ({xα}→ x0), then ℬ({xα})→ 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 x0 contains an element of ℬ({xα}), so ℬ({xα})→ x0.

15. Convergence of Associated Filters 2 X E1 E2 E3 E4 x1 x2 x3 x4

16. Construction of Associated Nets Proposition 3: Let ℬ = { Eα} α∊A be a filter base on a topological space X. Order A = {α} with the relation α ≤ βif Eα ⊇ Eβ. From each Eα select an arbitrary xα ∊ Eα. Then ж(ℬ) = {xα} 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, Eα ∩ Eβ ⊇E γ. Therefore α ≤ γ and β≤ γ, so A is directed. We now show that each xα ∊ X. Since each xα 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 x0 ∊ X, then any net associated with ℬ converges to x0. Proof: Let ж(ℬ) be a net associated with ℬ. Then for every neighborhood U of x0 ∃Eα ∊ ℬ such that Eα ⊆ U. Then ∀β≥ α, Eβ⊆Eα ⊆ U. Then ∀xβ ∊ Eβ, xβ ∊U. Therefore ж(ℬ)→ x0.

18. Convergence of Associated Nets X B1 B2 B3 B4 x1 x2 x3 x4

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