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单词 Filter
释义

filter


Let X be a set. A filter on X is a set 𝔽 of subsets of Xsuch that

  • X𝔽

  • The intersectionMathworldPlanetmath of any two elements of 𝔽 is an element of 𝔽.

  • 𝔽 (some authors do not include this axiom in the definition of filter)

  • If F𝔽 and FGX then G𝔽.

The first two axioms can be replaced by one:

  • Any finite intersection of elements of 𝔽 is an element of 𝔽.

with the usual understanding that the intersection of an empty familyof subsets of X is the whole set X.

A filter 𝔽 is said to be fixedor principal if there is F𝔽 such that no proper subsetMathworldPlanetmathPlanetmath of F belongs to 𝔽. In this case, 𝔽 consists of all subsets of X containing F, and F is called a principal element of 𝔽. If 𝔽 is not principal, it is said to be non-principal or free.

If x is any point (or any subset) of any topological spaceMathworldPlanetmath X,the set 𝒩x of neighbourhoods of x in X is a filter,called the neighbourhood filter of x.If 𝔽 is any filter on the space X, 𝔽is said to convergePlanetmathPlanetmath to x, and we write 𝔽x,if 𝒩x𝔽.If every neighbourhood of x meets every set of 𝔽, thenx is called an accumulation pointMathworldPlanetmathPlanetmathor cluster point of 𝔽.

Remarks: The notion of filter (due to H. Cartan) has a simplifying effect onvarious proofs in analysisMathworldPlanetmath and topology.TychonoffPlanetmathPlanetmath’s theorem would be one example.Also, the two kinds of limit that one sees in elementary realanalysis – the limit of a sequence at infinityMathworldPlanetmath, and the limitof a function at a point – are both special cases of the limitof a filter: the Fréchet filter and the neighbourhood filterrespectively.The notion of a Cauchy sequencePlanetmathPlanetmath can be extended with no difficultyto any uniform space (but not just a topological space),getting what is called a Cauchy filter; any convergent filter on auniform space is a Cauchy filter, and if the converseMathworldPlanetmath holds thenwe say that the uniform space is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Titlefilter
Canonical nameFilter
Date of creation2013-03-22 12:09:06
Last modified on2013-03-22 12:09:06
OwnerKoro (127)
Last modified byKoro (127)
Numerical id19
AuthorKoro (127)
Entry typeDefinition
Classificationmsc 03E99
Classificationmsc 54A99
Related topicUltrafilterMathworldPlanetmathPlanetmath
Related topicKappaComplete
Related topicKappaComplete2
Related topicNet
Related topicLimitAlongAFilter
Related topicUpperSet
Related topicOrderIdeal
Definesprincipal filter
Definesnonprincipal filter
Definesnon-principal filter
Definesfree filter
Definesfixed filter
Definesneighbourhood filter
Definesprincipal element
Definesconvergent filter
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更新时间:2025/5/4 20:58:29