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== | ||
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 <math>\mathcal{F}</math> with the property that for any subset <math>A \subseteq X</math> either <math>A \in \mathcal{F}</math> or the [[complement (set theory)|complement]] <math>X \setminus A \in \mathcal{F}</math>. | 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 <math>\mathcal{F}</math> with the property that for any subset <math>A \subseteq X</math> either <math>A \in \mathcal{F}</math> or the [[complement (set theory)|complement]] <math>X \setminus A \in \mathcal{F}</math>. | ||
The principal filter on a [[singleton]] set {''x''}, namely, all subsets of ''X'' containing ''x'', is an ultrafilter. | The principal filter on a [[singleton]] set {''x''}, namely, all subsets of ''X'' containing ''x'', is an ultrafilter.[[Category:Suggestion Bot Tag]] |
Latest revision as of 12:00, 16 August 2024
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.