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

1. F1 if F ∈ ℱ and G ⊇ F then G ∈ ℱ,

2. F2 if F ∈ ℱ and G ∈ ℱ then F ⋂ G ∈ ℱ,

3. F3 ø ∉ ℱ.

Here are three examples.

1. • If A is a non-empty subset of S then {F: A ⊆ F} is a filter.

2. • Suppose that (X, τ) is a topological space, and that x ∈ X. The collection Nx of neighbourhoods of x is a filter.

3. • 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.