In topology, the Nagata–Smirnov metrization theorem characterizes when a topological space is metrizable. The theorem states that a topological space is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (that is, -locally finite) basis.
A topological space is called a regular space if every non-empty closed subset of and a point p not contained in admit non-overlapping open neighborhoods. A collection in a space is countably locally finite (or -locally finite) if it is the union of a countable family of locally finite collections of subsets of
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.
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