 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.