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单词 UrysohnsLemma
释义

Urysohn’s lemma


A normal spaceMathworldPlanetmath is a topological spaceMathworldPlanetmath Xsuch that whenever A and B are disjoint closed subsets of X,then there are disjoint open subsets U and V of Xsuch that AU and BV.

(Note that some authors include T1 in the definition,which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to requiring the space to be HausdorffPlanetmathPlanetmath.)

Urysohn’s Lemma states that X is normalif and only ifwhenever A and B are disjoint closed subsets of X,then there is a continuous functionMathworldPlanetmathPlanetmath f:X[0,1]such that f(A){0} and f(B){1}.(Any such function is called an Urysohn function.)

A corollary of Urysohn’s Lemmais that normal T1 (http://planetmath.org/T1Space) spaces are completely regularPlanetmathPlanetmath.

TitleUrysohn’s lemma
Canonical nameUrysohnsLemma
Date of creation2013-03-22 12:12:34
Last modified on2013-03-22 12:12:34
Owneryark (2760)
Last modified byyark (2760)
Numerical id12
Authoryark (2760)
Entry typeTheorem
Classificationmsc 54D15
Related topicHowIsNormalityAndT4DefinedInBooks
Related topicApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces
DefinesUrysohn function
Definesnormal space
Definesnormal topological space
Definesnormal
DefinesnormalityPlanetmathPlanetmath
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