Difference between revisions of "Urysohn lemma"
From Encyclopedia of Mathematics
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| − | For any two disjoint closed sets | + | {{TEX|done}} |
| + | For any two disjoint closed sets $A$ and $B$ of a [[Normal space|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|Separation axiom]]; [[Urysohn–Brouwer lemma|Urysohn–Brouwer lemma]]). | ||
Latest revision as of 16:02, 19 April 2014
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).
Comments
The phrase "Urysohn lemma" is sometimes also used to refer to the Urysohn metrization theorem.
References
| [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 |
How to Cite This Entry:
Urysohn lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Urysohn_lemma&oldid=31873
Urysohn lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Urysohn_lemma&oldid=31873
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article