Urysohn's Metrization Theorem

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

Then $T$ is metrizable.

Proof
However, this form of the theorem was actually proved by Tychonoff in 1926.

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