# Metrization of Regular Second Countable Space

## Theorem

Let $T = \struct {S, \tau}$ be a $T_1$ space.

The following statements are equivalent:

$(1): \quad T$ is regular and second-countable
$(2): \quad T$ is homeomorphic to a metric subspace of the Hilbert cube $I^\omega$
$(3): \quad T$ is metrizable and separable

## Proof

### Condition $(1)$ implies Condition $(2)$

Follows immediately from Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube.

$\Box$

### Condition $(2)$ implies Condition $(3)$

Let $T$ be 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 $d_2$.

$T$ is homeomorphic to a metrizable space
$T$ is metrizable
the Hilbert cube $\struct{I^\omega, d_2}$ is separable
$T$ is homeomorphic to a separable space
$T$ is separable

$\Box$

### Condition $(3)$ implies Condition $(1)$

Let $T$ be metrizable and separable.

By definition of metrizable space:

there exists a metric $d$ on $S$ such that the topology $\tau$ on $S$ is the topology induced by $d$
$\struct{S, d}$ is a second-countable space
$\struct{S, \tau}$ is a fully normal space
$\struct{S, \tau}$ is a normal space
$\struct{S, \tau}$ is a regular space

$\blacksquare$

## Also known as

In some sources this theorem is known as Urysohn's Metrization Theorem as it includes the sufficient condition to the necessary conditions of metrization that is the subject of what is more commonly known as Urysohn's Metrization Theorem.