Urysohn’s lemma

A normal spaceMathworldPlanetmath is a topological spaceMathworldPlanetmath X such that whenever A and B are disjoint closed subsets of X, then there are disjoint open subsets U and V of X such 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 normal if and only if whenever 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 Lemma is that normal T1 (http://planetmath.org/T1Space) spaces are completely regularPlanetmathPlanetmath.

Title Urysohn’s lemma
Canonical name UrysohnsLemma
Date of creation 2013-03-22 12:12:34
Last modified on 2013-03-22 12:12:34
Owner yark (2760)
Last modified by yark (2760)
Numerical id 12
Author yark (2760)
Entry type Theorem
Classification msc 54D15
Related topic HowIsNormalityAndT4DefinedInBooks
Related topic ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces
Defines Urysohn function
Defines normal space
Defines normal topological space
Defines normal
Defines normalityPlanetmathPlanetmath