# Filter (mathematics)

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

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

1. 
2. 
3. 
4. 

If G is a subset of X then the family



is a filter, the principal filter on 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 on a singleton set {x}, namely, all subsets of X containing x, is an ultrafilter.