# Urysohn metrization theorem

From Encyclopedia of Mathematics

A compact or countably compact Hausdorff space is metrizable if and only if it has a countable base: indeed, it is homeomorphic to a subset of the Hilbert cube.

A topological space with a countable base is metrizable if and only if it is normal, or (an addition by A.N. Tikhonov) if and only if it is regular.

#### References

[a1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. Chapt. 5 (Translated from Russian) |

[a2] | J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 125; 127 |

[a3] | W.Franz, "General topology" , Harrap (1967) p. 100 |

**How to Cite This Entry:**

Urysohn metrization theorem.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Urysohn_metrization_theorem&oldid=33673

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