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单词 Ultrafilter
释义

ultrafilter


Let X be a set.

Definitions

A collectionMathworldPlanetmath 𝒰 of subsets of X is an ultrafilter if 𝒰 is a filter, and whenever AX then either A𝒰 or XA𝒰.

Equivalently, an ultrafilter on X is a maximal (http://planetmath.org/MaximalElement) filter on X.

More generally, an ultrafilter of a latticeMathworldPlanetmath (http://planetmath.org/LatticeFilter)is a maximal proper filter of the lattice.This is indeed a generalizationPlanetmathPlanetmath, as an ultrafilter on Xcan then be defined as an ultrafilter of the power setMathworldPlanetmath 𝒫(X).

Types of ultrafilter

For any xX the set {AXxA} is an ultrafilter on X.An ultrafilter formed in this way is called a fixed ultrafilter,or a principal ultrafilter, or a trivial ultrafilter.Any other ultrafilter on X is called a free ultrafilter,or a non-principal ultrafilter.An ultrafilter on a finite setMathworldPlanetmath is necessarily fixed.On any infinite setMathworldPlanetmath there are free ultrafilters(in great abundance (http://planetmath.org/NumberOfUltrafilters)),but their existence depends on the Axiom of ChoiceMathworldPlanetmath,and so none can be explicitly constructed.

An ultrafilter 𝒰 on X is called a uniform ultrafilterif every member of 𝒰 has the same cardinality.(An ultrafilter on a singleton is uniform,but this is a degenerate case and is often excluded.All other uniform ultrafilters are free.)

Titleultrafilter
Canonical nameUltrafilter
Date of creation2013-03-22 12:13:59
Last modified on2013-03-22 12:13:59
Owneryark (2760)
Last modified byyark (2760)
Numerical id11
Authoryark (2760)
Entry typeDefinition
Classificationmsc 54A20
Related topicFilter
Related topicUltranet
Related topicEveryBoundedSequenceHasLimitAlongAnUltrafilter
Related topicLatticeFilter
Definesfixed ultrafilter
Definesprincipal ultrafilter
Definestrivial ultrafilter
Definesfree ultrafilter
Definesnon-principal ultrafilter
Definesnonprincipal ultrafilter
Definesuniform ultrafilter
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更新时间:2025/5/4 4:09:34