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Revision difference : Urysohn's lemma |
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| \PMlinkescapeword{corollary} |
Let $X$ be a normal topological space, and let $C,D\subseteq X$ be disjoint closed nonempty subsets. Then there is a continuous function $f:X\rightarrow [0,1]$ such that $f(C) = \{ 0\}$ and $f(D)= \{ 1\}$. |
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| Let $X$ be a normal topological space, and let $C,D\subseteq X$ be disjoint closed nonempty subsets. Then there is a continuous function $f\colon X\rightarrow [0,1]$ such that $f(C) = \{0\}$ and $f(D) = \{1\}$. |
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| (Any such function is called an \emph{Urysohn function}.) |
(Any such function is called an \emph{Urysohn function}.) |
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| A corollary is that normal Hausdorff spaces are completely regular. |
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