Urysohn's lemma UrysohnsLemma 2002-01-22 12:30:30 2007-05-23 13:12:02 Theorem Urysohn function normal space normal topological space normal normality topology \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{corollary} A \emph{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.) \emph{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 \emph{Urysohn function}.) A corollary of Urysohn's Lemma is that normal \PMlinkname{$\mathrm{T}_1$}{T1Space} spaces are completely regular.