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# Mathematical filter

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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 Fréchet filter on an infinite set S. This is the set of all subsets of S which have finite complement.

Filters are useful in topology. For example, 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.

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Last edited August 15, 2001 11:46 am (diff)
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