Urysohn metrization theorem

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A compact or countably compact Hausdorff space is metrizable if and only if it has a countable base.

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



[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
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Urysohn metrization theorem. 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