Tychonoff's theorem: the product of compact spaces is compact — via ultrafilters
Filter Controls
A filter on a set X is a collection F of subsets closed under supersets and finite intersections, with ∅ ∉ F.
An ultrafilter is maximal: for every A ⊆ X, either A ∈ F or X\A ∈ F. By Zorn's lemma, every filter extends to an ultrafilter.
Product Space [0,1]²
Tychonoff's Theorem: For compact spaces {X_α}, the product ∏X_α is compact.
Proof via ultrafilters: any ultrafilter on the product has a convergent subnet (projections converge in each factor by compactness), so the product is compact.