Difference between revisions of "Filter (mathematics)"

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form a filter, the ''neighbourhood filter'' of ''x''.
 
form a filter, the ''neighbourhood filter'' of ''x''.
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===Filter bases===
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A '''base''' <math>\mathcal{B}</math> for the filter <math>\mathcal{F}</math> is a non-empty collection of non-empty sets such that the family of subsets of ''X'' containing some element of <math>\mathcal{B}</math> is precisely the filter <math>\mathcal{F}</math>.
  
 
==Ultrafilters==
 
==Ultrafilters==

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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:

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.