# Urysohn's Metrization Theorem

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 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

It follows that $T$ is metrizable by definition.

$\blacksquare$

## 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.

## 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.

## Sources

- 1955: John L. Kelley:
*General Topology*: Chapter $4$: Embedding and Metrization, $\S$ Metrization, Theorem $16$ - 1970: Stephen Willard:
*General Topology*: Chapter $7$: Metrizable Spaces: $\S23$: Metrization: Definition $23.1$ - 1975: James R. Munkres:
*Topology*: Chapter $4$: Countability and Separation Axioms: $\S34$: The Urysohn Metrization Theorem: Theorem $34.1$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metrizability