A Topological space is metrizable if there is a Metric such that the Metric topology with respect to that metric equals the topology in the space. vid.

An important problem in topology is finding necessary and sufficient conditions for metrizability

A discrete space is metrizable (vid).

A metrizable space is Hausdorff

Example: R^n, with metric topology with Euclidean metric. It equals the Product topology (using Standard topology for $\mathbb{R}$.

Is $\mathbb{R}^\omega$ (the space of infinite sequences of real numbers), and more generally $\mathbb{R}^J$, for arbitrary Index set $J$, metrizable? More precisely, is there a topology on this space which is metrizable? Yes, with the Uniform metric

If X is metrizable and has a countable dense subset D then X has a countable basis (it is Second countable).

A metrizable topological space with corresponding metric, is called metric space

Necessary and sufficient conditions

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Necessary: Hausdorff, first countable, normal

Sufficient: second countable & normal (Urysohn metrization theorem).

–> There are more general Metrization theorems showing a general set of necessary and sufficient conditions (which includes normality)