Nagata–Smirnov metrization theorem
Template:Short description In topology, the Nagata–Smirnov metrization theorem characterizes when a topological space is metrizable. The theorem states that a topological space <math>X</math> is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (that is, Template:Sigma-locally finite) basis.
A topological space <math>X</math> is called a regular space if every non-empty closed subset <math>C</math> of <math>X</math> and a point p not contained in <math>C</math> admit non-overlapping open neighborhoods. A collection in a space <math>X</math> is countably locally finite (or Template:Sigma-locally finite) if it is the union of a countable family of locally finite collections of subsets of <math>X.</math>
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<ref>J. Nagata, "On a necessary and sufficient condition of metrizability", J. Inst. Polytech. Osaka City Univ. Ser. A. 1 (1950), 93–100.</ref> and 1951,<ref>Y. Smirnov, "A necessary and sufficient condition for metrizability of a topological space" (Russian), Dokl. Akad. Nauk SSSR 77 (1951), 197–200.</ref> respectively.