Filter (mathematics)

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 that include a particular subset C of S. Such a filter is called the "principal filter" generated by C. The Fréchet filter on an infinite set S is the set of all subsets of S that 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. A standard application of Zorn's lemma shows that every filter is a subset of some ultrafilter.

For any filter F on a set S, the set function defined by

is finitely additive -- a "measure" if that term is construed rather loosely. Therefore the statement

can be considered somewhat analogous to the statement that φ holds "almost everywhere". That interpretation of membership in a filter is used (for motivation, although it is not needed for actual proofs) in the theory of ultraproducts in model theory, a branch of mathematical logic.

Filters as described above are subsets of a particular lattice, namely the power set of S. Filters can be also be defined in other lattices - see lattice for details.