Metrization theorems

A metrizable space is a topological space which is homeomorphic to a metric space. Metrization theorems are theorems which give sufficient conditions for a topological space to be metrizable.

For explanations of many of the terms used in this article, the reader should see the article on topology.

The first really useful metrization theorem was Urysohn's Metrization Theorem. This states that every second-countable regular Hausdorff space is metrizable. So, for example, every second-countable manifold is metrizable.

Several other metrization theorems follow as simple corollaries to Urysohn's Theorem. For example, a compact Hausdorff space is metrizable if and only if it is second-countable.

Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is second-countable, regular and Hausdorff. The Nagata-Smirnov Metrization Theorem extends this to the non-separable case. It states that a topological space is metrizable if and only if it is regular and Hausdorff and has a σ-locally finite base. A σ-locally finite base is a base which is a union of countably many locally finite collections of open sets.

A space is said to be locally metrizable if every point has a metrizable neighbourhood. Smirnov proved that a locally metrizable Hausdorff space is metrizable if and only if it is paracompact. In particular, a manifold is metrizable if and only if it is paracompact.