# Metrization of Regular Second Countable Space

This article needs proofreading.Please check it for mathematical errors.If you believe there are none, please remove `{{Proofread}}` from the code.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Proofread}}` from the code. |

## 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 $d_2$.

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

From Hilbert Cube is Separable:

- the Hilbert cube $\struct{I^\omega, d_2}$ is separable

From Subspace of Separable Metric Space is Separable:

- $T$ is homeomorphic to a separable space

From Continuous Image of Separable Space is Separable:

- $T$ 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 $d$

From Metric Space is Separable iff Second-Countable

- $\struct{S, d}$ is a second-countable space

From Metric Space is Fully Normal:

- $\struct{S, \tau}$ is a fully normal space

From Fully Normal Space is Normal Space:

- $\struct{S, \tau}$ is a normal space

From Normal Space is Regular 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.

## Also see

## Sources

- 1955: John L. Kelley:
*General Topology*: Chapter $4$: Embedding and Metrization - 1970: Stephen Willard:
*General Topology*: Chapter $7$: Metrizable Spaces: $\S23$: Metrization: Definition $23.1$