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 (X,τ) is said to be metrizable if there is a metric d\colon X \times X \to [0,\infty) such that the topology induced by d is τ. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.

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Properties

Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, this is not generally true for topological spaces with additional structure. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic.

Metrization theorems

The first really useful metrization theorem was Urysohn's metrization theorem. This states that every second-countable regular 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 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 regular, Hausdorff and second-countable. 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, 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. For a closely related theorem see the Bing metrization theorem.

Separable metrizable spaces can also be characterized as those spaces which are homeomorphic to a subspace of the Hilbert cube \lbrack 0,1\rbrack ^\mathbb{N}, i.e. the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the product topology.

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