# Filter (mathematics)

In set theory, a **filter** is a family of subsets of a given set which has properties generalising those of neighbourhoods in topology.

Formally, a filter on a set *X* is a subset of the power set with the properties:

If *G* is a nonempty subset of *X* then the family

is a filter, the *principal filter* generated by *G*.

In a topological space , the neighbourhoods of a point *x*

form a filter, the *neighbourhood filter* of *x*.

### Filter bases

A **base** for the filter is a non-empty collection of non-empty sets such that the family of subsets of *X* containing some element of is precisely the filter .

## Ultrafilters

An **ultrafilter** is a maximal filter: that is, a filter on a set which is not properly contained in any other filter on the set. Equivalently, it is a filter with the property that for any subset either or the complement .

The principal filter generated by a singleton set {*x*}, namely, all subsets of *X* containing *x*, is an ultrafilter.

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