Nagata–Smirnov metrization theorem

In topology, the Nagata–Smirnov metrization theorem characterizes when a topological space is metrizable. The theorem states that a topological space $$X$$ is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (that is, $\sigma$-locally finite) basis.

A topological space $$X$$ is called a regular space if every non-empty closed subset $$C$$ of $$X$$ and a point p not contained in $$C$$ admit non-overlapping open neighborhoods. A collection in a space $$X$$ is countably locally finite (or 𝜎-locally finite) if it is the union of a countable family of locally finite collections of subsets of $$X.$$

Unlike Urysohn's metrization theorem, which provides only a sufficient condition for metrizability, this theorem provides both a necessary and sufficient condition for a topological space to be metrizable. The theorem is named after Junichi Nagata and Yuriĭ Mikhaĭlovich Smirnov, whose (independent) proofs were published in 1950 and 1951, respectively.