# Urysohn lemma

From Encyclopedia of Mathematics

For any two disjoint closed sets and of a normal space there exists a real-valued function , continuous at all points, taking the value at all points of , the value 1 at all points of and for all satisfying the inequality . This lemma expresses a condition which is not only necessary but also sufficient for a -space 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=15326

This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article