Nagata–Smirnov metrization theorem
The Nagata–Smirnov metrization theorem in topology 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 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 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 proofs were published in 1950 and 1951, respectively.
