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Filter (mathematics) Redirected from Mathematical filter
A filter F on a set S is a set of subsets of S with the following properties:
- S is in F.
- The empty set is not in F.
- If A and B are in F, then so is their intersection.
- 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.
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