Filters in topology

In mathematics, a filter on a set is a collection of nonempty subsets which is upward closed and closed under finite intersections (see Filter on a set for a more detailed discussion). An example of a filter is the collection of neighborhoods of a point in a topological space. This article is focused on applications of filters to topology.

Filters were introduced by Henri Cartan in 1937 as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. In metric spaces, and more generally in sequential spaces, basic topological notions such as open set, closed set, convergence, continuity, compactness and more can be fully characterized in terms of sequences, as studied in real analysis. But in other classes of spaces, sequences are insufficient. Filters and nets both provide ways to generalize the notion of convergence and characterize all these topological notions in general topological spaces. Filters also provide a common framework for defining various types of limits of functions, including limits from the left or right, to infinity, to a point or a set.

In particular, filters and nets provide equivalent notions of convergence: to each net one can associate a filter, and to each filter a net, so that the net converges to a point if and only if the same is true of the associated filter, and vice versa. So the use of one or the other is often a matter of convenience.

Besides characterizing topological notions, filters and ultrafilters, which are a special type of filter, are used extensively in general topology for constructing topological spaces of interest.