Metrization theorem

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In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space <math>(X,\tau)<math> is said to be metrizable if there is a metric <math>d\colon X\to [0,\infty)<math> such that the topology induced by d is <math>\tau<math>. Metrization theorems are theorems that 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 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 second-countable regular Hausdorff space is metrizable. So, for example, every second-countable 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 second-countable 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 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.

See also

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