Urysohn lemma

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For any two disjoint closed sets $A$ and $B$ of a normal space $X$ there exists a real-valued function $f$, continuous at all points, taking the value $0$ at all points of $A$, the value 1 at all points of $B$ and for all $x\in X$ satisfying the inequality $0\leq f(x)\leq1$. This lemma expresses a condition which is not only necessary but also sufficient for a $T_1$-space $X$ to be normal (cf. also Separation axiom; Urysohn–Brouwer lemma).


The phrase "Urysohn lemma" is sometimes also used to refer to the Urysohn metrization theorem.


[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 123–124 (Translated from Russian)
[a2] J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 115
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Urysohn lemma. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article