**Appendix D**

Tychonoff’s theorem

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 ...

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