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# Contents

## Definition

A topological space $(X,\tau)$ is called metrisable if there exists the stucture of a metric space $(X,d)$ on the underlying set, such that $\tau$ is the corresponding metric topology. If there exists such a $(X,d)$ which is complete, then $(X,\tau)$ is called completely metrisable.

## Properties

Metrizable spaces enjoy a number of separation properties: they are Hausdorff, regular, and even normal. They are also paracompact.

Metrizable spaces are closed under topological coproducts and of course subspaces (and therefore equalizers); they are closed under countable products but not general products (for instance, a product of uncountably many copies of the real line $\mathbb{R}$ is not a normal space).

## Metrisability theorem

Fundamental early work in point-set topology established a number of metrization theorems, i.e., theorems which give sufficient conditions for a space to be metrisable. One of the more useful theorems is Urysohn metrization theorem: A regular, Hausdorff, and second-countable space is metrisable. So, for instance, a compact Hausdorff space that is second-countable is metrisable. Other metrization theorems are:

• Nagata-Smirnov metrization theorem

• Bing metrization theorem

• Moore metrization theorem