= Ultrafilter characterization of compactness
A <topological space> is <compact> if and only if every <ultrafilter> on it converges to at least one point, assuming the <ultrafilter lemma>. No <Hausdorff> assumption is needed. In a compact space, the closures of the filter members have the <finite intersection property>, so choose a point in their intersection. Every open neighbourhood of that point belongs to the ultrafilter, since otherwise its closed complement would be a filter member. Conversely, extend the complements of a cover with no finite subcover to an ultrafilter; a limit point lies in a covering open set, contradicting the simultaneous membership of that set and its complement.
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