Ultralimit

A point $x$ is the limit of a map $f:I\to X$ along an ultrafilter $\mathcal{U}$ on $I$ when $f^{-1}(V)\in \mathcal{U}$ for every neighbourhood $V$ of $x$. Every ultrafilter on a compact space has a limit, and that limit is unique in a Hausdorff space.

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.