Compactness in terms of convergence of ultra filters I'm reading this article on Matheplanet. I have trouble with the following theorem:

Satz 5 : Ein topologischer Raum X ist genau dann kompakt wenn alle Ultrafilter auf X konvergieren


Translation : A topological space $X$ is compact exactly when every ultra filter on $X$ converges.

How can an ultra filter converge? I thought ultra filter was like a collection of subsets, and as far as I know, there is no idea of convergence of "subsets" in topology.

By definition, a filter $\mathcal{F}$ converges to a point $x$ iff $\mathcal{F}$ contains all neighbourhoods of $x$.
