###### Theorem

A topological space $X$ is metrisable if and only if it is regular, Hausdorff and has a countably locally finite base.

Nagata-Smirnov metrization theorem

The Nagata-Smirnov metrization theorem gives a necessary and sufficient condition under which a space is metrisable.

A topological space $X$ is metrisable if and only if it is regular, Hausdorff and has a countably locally finite base.

- Nagataâ€“Smirnov metrization theorem, Wikipedia

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