Mathematical filter

A filter F on a set S is a set of subsets of S with the following properties:

1. S is in F.
2. The empty set is not in F.
3. If A and B are in F, then so is their intersection.
4. If A is in F and A ⊆ B ⊆ S, then B is in F.

A simple example of a filter is the set of all subsets of S which contain a common point x. The Fréchet filter on an infinite set S is the set of all subsets of S which have finite complement.

Filters are useful in topology: they play the role of sequences in metric spaces. The set of all neighbourhoods of a point x in a topological space is a filter, called the neighbourhood filter of x. A filter which is a superset of the neighbourhood filter of x is said to converge to x. Note that in a non-Hausdorff space a filter can converge to more than one point.

Of particular importance are maximal filters, which are called ultrafilters.