The power of Nets! Tychonoff Theorem Product spaces When - PowerPoint PPT Presentation

The power of Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets The power of Nets! Tychonoff Theorem Product spaces When sequences are not enough The Theorem Summary Jaspreet Kaur Directed Reading Program, 2015

The power of Outline Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff Theorem 1 Nets and Ultranets Product spaces The Theorem Nets Summary Ultranets 2 Tychonoff Theorem Product spaces The Theorem

The power of Nets Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff Theorem Product spaces The Theorem Definition Summary A net is a function P : Λ → X , where Λ is a directed set and X is an arbitrary set. We denote P ( λ ) by x λ , and the net by ( x λ ) λ ∈ Λ .

The power of Directed sets Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Definition Tychonoff Theorem A set Λ is said to be a direcrted set if there is a relation ≤ on Λ Product spaces such that the following hold The Theorem Summary

The power of Directed sets Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Definition Tychonoff Theorem A set Λ is said to be a direcrted set if there is a relation ≤ on Λ Product spaces such that the following hold The Theorem Summary 1 The relation is transitive, i.e. if λ 1 ≤ λ 2 and λ 2 ≤ λ 3 , then λ 1 ≤ λ 3 ; and

The power of Directed sets Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Definition Tychonoff Theorem A set Λ is said to be a direcrted set if there is a relation ≤ on Λ Product spaces such that the following hold The Theorem Summary 1 The relation is transitive, i.e. if λ 1 ≤ λ 2 and λ 2 ≤ λ 3 , then λ 1 ≤ λ 3 ; and 2 If λ 1 and λ 2 are in Λ, then there is a λ 3 so that λ 1 ≤ λ 3 and λ 2 ≤ λ 3

The power of Directed sets Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Definition Tychonoff Theorem A set Λ is said to be a direcrted set if there is a relation ≤ on Λ Product spaces such that the following hold The Theorem Summary 1 The relation is transitive, i.e. if λ 1 ≤ λ 2 and λ 2 ≤ λ 3 , then λ 1 ≤ λ 3 ; and 2 If λ 1 and λ 2 are in Λ, then there is a λ 3 so that λ 1 ≤ λ 3 and λ 2 ≤ λ 3 Example The natural numbers with their usual order operation.

The power of Ultranets Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Definition Tychonoff A net is said to be an ultranet if for every subset A of X , Theorem Product spaces ( x λ ) λ ∈ Λ is eventually in A or eventually in X \ A . The Theorem Summary

The power of Ultranets Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Definition Tychonoff A net is said to be an ultranet if for every subset A of X , Theorem Product spaces ( x λ ) λ ∈ Λ is eventually in A or eventually in X \ A . The Theorem Summary Definition A net ( x λ ) λ ∈ Λ in X is said to be eventually in A ⊂ X if there is a λ 0 ∈ Λ so that for every λ ≥ λ 0 , x λ ∈ A .

The power of Ultranets Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Definition Tychonoff A net is said to be an ultranet if for every subset A of X , Theorem Product spaces ( x λ ) λ ∈ Λ is eventually in A or eventually in X \ A . The Theorem Summary Definition A net ( x λ ) λ ∈ Λ in X is said to be eventually in A ⊂ X if there is a λ 0 ∈ Λ so that for every λ ≥ λ 0 , x λ ∈ A . Definition A net ( x λ ) in X converges to x ∈ X if for each neighborhood U of x , there is a λ 0 ∈ Λ so that λ ≥ λ 0 implies x λ is in U . Equivalently, the net is enventually in every neighborhood of x .

The power of Product spaces Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff Theorem Product spaces The Theorem Summary Theorem A net ( x λ ) in a product � α ∈ Γ X α converges to x ∈ X if and only if for each α ∈ Γ , J α ( x λ ) → J α ( x ) in X α .

The power of Product spaces Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff Theorem Product spaces The Theorem Summary Theorem A net ( x λ ) in a product � α ∈ Γ X α converges to x ∈ X if and only if for each α ∈ Γ , J α ( x λ ) → J α ( x ) in X α .

The power of Compact spaces Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff For many spaces sequences are enough to characterize Theorem Product spaces compactness; this is usually presented as sequential The Theorem compactness is equivalent to compactness in most ”nice” Summary spaces. There are counterexamples to that statement, however,in more general settings. All is not lost, as can be seen from the next theorem.

The power of Compact spaces Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff For many spaces sequences are enough to characterize Theorem Product spaces compactness; this is usually presented as sequential The Theorem compactness is equivalent to compactness in most ”nice” Summary spaces. There are counterexamples to that statement, however,in more general settings. All is not lost, as can be seen from the next theorem. Theorem A space X is compact if and only if every ultranet converges in X.

The power of Continuous maps Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets To prove the result of the Tychonoff Theorem we need two Tychonoff Theorem more lemmas in addition to the above Theorems. Product spaces The Theorem Summary

The power of Continuous maps Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets To prove the result of the Tychonoff Theorem we need two Tychonoff Theorem more lemmas in addition to the above Theorems. Product spaces The Theorem Lemma Summary The projection maps J α : � α ∈ Γ X α → X α , are continuous for each α .

The power of Continuous maps Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets To prove the result of the Tychonoff Theorem we need two Tychonoff Theorem more lemmas in addition to the above Theorems. Product spaces The Theorem Lemma Summary The projection maps J α : � α ∈ Γ X α → X α , are continuous for each α . Lemma The continuous image of a compact space is compact.

The power of Continuous maps Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets To prove the result of the Tychonoff Theorem we need two Tychonoff Theorem more lemmas in addition to the above Theorems. Product spaces The Theorem Lemma Summary The projection maps J α : � α ∈ Γ X α → X α , are continuous for each α . Lemma The continuous image of a compact space is compact. Each of the above hold in general, and do not require the use of nets.

The power of Tychonoff Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff Theorem (Tychonoff) Theorem Product spaces The non-empty product X = � α ∈ Γ X α is compact if and only if The Theorem Summary each factor is compact. The original proof of this theorem did not use nets and is much harder to prove. The proof we give below hides this difficulty in the notion of ultranets, and is only four lines.

The power of Proof of Tychonoff Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff Proof. Theorem Product spaces The forward direction is a consequence of two previous lemmas The Theorem and the assumption that the product is compact. Summary

The power of Proof of Tychonoff Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff Proof. Theorem Product spaces The forward direction is a consequence of two previous lemmas The Theorem and the assumption that the product is compact. Summary Now assume that each factor, X α , is compact and let ( x λ ) λ ∈ Λ be an ultranet in X = � α ∈ Γ X α . Then for each α , ( J α ( x λ )) is an ultranet in X α and hence converges, as each factor is compact. This says that ( x λ ) converges in X by the previous theorem. Finally, X is compact since every ultranet in X converges.

The power of Summary Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff Theorem • Nets take on the role of sequences when the spaces Product spaces The Theorem become more complicated, e.q. spaces without a metric. Summary

The power of Summary Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff Theorem • Nets take on the role of sequences when the spaces Product spaces The Theorem become more complicated, e.q. spaces without a metric. Summary • The notion of an ultra-net characterizes compactness in a more general setting that sequential compactness can account for.

The power of Summary Nets! Jaspreet Kaur Nets and Ultranets Nets Ultranets Tychonoff Theorem • Nets take on the role of sequences when the spaces Product spaces The Theorem become more complicated, e.q. spaces without a metric. Summary • The notion of an ultra-net characterizes compactness in a more general setting that sequential compactness can account for. • The Tychonoff Theorem states that an arbitrary product of compact spaces is again compact.

The power of For Further Reading I Nets! Jaspreet Kaur Appendix For Further Reading S. Willard. General Topology . Dover Books, 1970.