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 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 firstcountable. 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 secondcountable regular 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 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 regular, Hausdorff and secondcountable. 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, 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. 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 , 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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