Filter (mathematics): Difference between revisions
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form a filter, the ''neighbourhood filter'' of ''x''. | form a filter, the ''neighbourhood filter'' of ''x''. | ||
===Filter bases=== | |||
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== |
Revision as of 17:47, 1 December 2008
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.