# Urysohn’s lemma

A normal space is a topological space $X$ 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 $A\subseteq U$ and $B\subseteq V$.

(Note that some authors include $\mathrm{T}_{1}$ in the definition, which is equivalent to requiring the space to be Hausdorff.)

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 function $f\colon X\to[0,1]$ such that $f(A)\subseteq\{0\}$ and $f(B)\subseteq\{1\}$. (Any such function is called an Urysohn function.)

A corollary of Urysohn’s Lemma is that normal $\mathrm{T}_{1}$ (http://planetmath.org/T1Space) spaces are completely regular.

 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 normality