Urysohn's Metrization Theorem

Theorem
Let $T = \struct {S, \tau}$ be a topological space which is regular and second-countable.

Then $T$ is metrizable.

Proof
From Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube:
 * $T$ is homeomorphic to a subspace of the Hilbert cube $\struct{I^\omega, d_2}$

where $d_2$ is the metric: $\ds \forall x = \sequence {x_i}, y = \sequence {y_i} \in I^\omega: \map {d_2} {x, y} := \paren {\sum_{k \mathop \in \N_{>0} } \paren {x_k - y_k}^2}^{\frac 1 2}$

By definition of metrizable topology:
 * $\struct{I^\omega, \tau_{d_2}}$ is metrizable

where $\tau_{d_2}$ is the topology induced by $d_2$.

From Subspace of Metrizable Space is Metrizable Space:
 * $T$ is homeomorphic to a metrizable space

From Topological Space Homeomorphic to Metrizable Space is Metrizable Space:
 * $T$ is metrizable

Also see

 * Metrizable Space is not necessarily Second-Countable, indicating that the converse does not hold.


 * Metrization of Regular Second Countable Space, for necessary and sufficient conditions for the metrization of regular second-countable spaces.