# Urysohn's Metrization Theorem

## Theorem

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

Then $T$ is metrizable.

## 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.