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 nonHausdorff 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
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