= Ultrafilter characterization of compact Hausdorff spaces
{title2=$\forall\mathcal U\ \exists!\,x:\mathcal U\to x$}
A <topological space> is both a <compact space> and a <Hausdorff space> exactly when every <ultrafilter> on its underlying <set> converges to exactly one point. Existence of all ultrafilter limits characterizes compactness. If a space is not Hausdorff, two distinct points have neighbourhood systems whose combined finite intersections are nonempty; the <ultrafilter lemma> extends these to an ultrafilter converging to both points. Conversely, disjoint neighbourhoods prevent such a common limit.
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