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

<math>

m(A)=\left\{ \begin{matrix} \,1 & \mbox{if }A\in F \\ \,0 & \mbox{if }S\setminus A\in F \\ \,\mbox{undefined} & \mbox{otherwise} \end{matrix} \right. </math> is finitely additive -- a "measure" if that term is construed rather loosely. Therefore the statement

<math>\left\{\,x\in S: \varphi(x)\,\right\}\in F</math>

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.