For explanations of many of the terms used in this article, the reader should see the Topology Glossary.
Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first countable.
The first really useful metrization theorem was Urysohn's Metrization Theorem. This states that every secondcountable regular Hausdorff space is metrizable. So, for example, every secondcountable manifold is metrizable. (Historical note: The form of the theorem shown here was in fact proved by Tychonoff[?] in 1926. What Urysohn had shown, in a paper published posthumously in 1925, was the slightly weaker result that every secondcountable normal Hausdorff space 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 secondcountable.
Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is secondcountable, regular and Hausdorff. The NagataSmirnov Metrization Theorem extends this to the nonseparable case. It states that a topological space is metrizable if and only if it is regular and Hausdorff and has a σlocally finite base. A σlocally 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.
Search Encyclopedia

Featured Article
