Encyclopedia > Filter (mathematics)

  Article Content

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 ABS, 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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Raymond IV of Toulouse

... to put difficulties in the way of Bohemund's retention of Antioch, obstinately alleging the oath to Alexius, and refusing to surrender the positions in the city which he had ...