Metrizable

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 ${\displaystyle (X,{\mathcal {T}})}$ is said to be metrizable if there is a metric

such that the topology induced by d is ${\displaystyle {\mathcal {T}}}$.Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.

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, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic.

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