Urysohn's Metrization Theorem

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space which is regular and second-countable.


Then $T$ is metrizable.


Proof


Also see


Source of Name

This entry was named for Pavel Samuilovich Urysohn.


Historical Note

This form of Urysohn's Metrization Theorem was actually proved by Andrey Nikolayevich Tychonoff in 1926.

What Urysohn had shown, in a posthumous $1925$ paper, was that every second-countable normal Hausdorff space is metrizable.


Sources