metrisable topological space in nLab

Contents

Contents

Definition

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.

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 ℝ\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

• metric space

• separable metric space

• topological space

• Polish space

References

See also

• Wikipedia, Metrization theorem