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 $f:X\to [0,1]$ 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