Filter

Summary

A filter on a set $X$ is a nonempty family of subsets of $X$ that does not contain $\emptyset$. It contains every superset of its members and the intersection of any two of its members. One thinks of the members of a filter as the "large" sets, or as the sets that contain "almost all" points in some sense. Filters give a theory of convergence that works in every topological space, where sequences are not enough. Maximal filters are the ultrafilters.