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.

Metrizable spaces are closed under topological coproducts and of course subspaces (and therefore equalizers); they are closed under countableproducts 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 Urysohnmetrization 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: