Urysohn metrization theorem
From HandWiki
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 |
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