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 of subsets of a set S is a filter if
F1 if F ∈ and G ⊇ F then G ∈ ,
F2 if F ∈ and G ∈ then F ∩ G ∈ ,
F3 ∉ .
Here are three examples.
•If A is a non-empty subset of ...