nLab metrisable topological space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



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


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


See also

Last revised on March 20, 2024 at 20:19:19. See the history of this page for a list of all contributions to it.