Appendix D - Tychonoff's theorem
Published online by Cambridge University Press: 05 June 2014
Summary
We prove Tychonoff's theorem, that the topological product of compact topological spaces is compact. The key idea is that of a filter. This generalizes the notion of a sequence in a way which allows the axiom of choice to be applied easily.
A collection ℱ of subsets of a set S is a filter if
F1 if F ∈ ℱ and G ⊇ F then G ∈ ℱ,
F2 if F ∈ ℱ and G ∈ ℱ then F ⋂ G ∈ ℱ,
F3 ø ∉ ℱ.
Here are three examples.
• If A is a non-empty subset of S then {F: A ⊆ F} is a filter.
• Suppose that (X, τ) is a topological space, and that x ∈ X. The collection Nx of neighbourhoods of x is a filter.
• If (sn) is a sequence in S then
is a filter.
Filters can be ordered. We say that G refines ℱ, and write G ≥ ℱ, if G ⊇ ℱ.
We now consider a topological space (X, τ). We say that a filter ℱ converges to a limit x (and write ℱ → x) if ℱ refines Nx. Clearly if G refines ℱ and ℱ → x then G → x.
The Hausdorff property can be characterized in terms of convergent filters.
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- Information
- A Course in Mathematical Analysis , pp. 607 - 611Publisher: Cambridge University PressPrint publication year: 2014