Convergence of a filter ($\mathcal{F}⟶x$)

A filter on a set equipped with a topology converges to $x$ if it contains every neighbourhood of $x$. A Hausdorff space gives every convergent filter at most one limit: disjoint neighbourhoods of distinct purported limits would have empty intersection in the filter.

A continuous dense map into a compact Hausdorff space assigns to each ultrafilter on its domain the unique limit of its pushforward. This limit map is surjective by the ultrafilter characterization of image closure. It is injective exactly when disjoint domain subsets have disjoint image closures.