In topology, the Nagata–Smirnov metrization theorem characterizes when a topological space is metrizable. The theorem states that a topological space X {\displaystyle X} is metrizable if and only if it is regular and has a countably locally finite (that is, 𝜎-locally finite) basis. A topological space X {\displaystyle X} is called a regular space if every non-empty closed subset C {\displaystyle C} of X {\displaystyle X} and a point p {\displaystyle p} not contained in C {\displaystyle C} admit non-overlapping open neighborhoods. A collection in a space X {\displaystyle 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 . {\displaystyle 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.
See also Bing metrization theorem – Characterizes when a topological space is metrizable Kolmogorov's normability criterion – Characterization of normable spaces Uniformizable space – Topological space whose topology is generated by a uniform structure
Notes
References Munkres, James R. (1975), "Sections 6-2 and 6-3", Topology, Prentice Hall, pp. 247–253, ISBN 0-13-925495-1. Patty, C. Wayne (2009), "7.3 The Nagata–Smirnov Metrization Theorem", Foundations of Topology (2nd ed.), Jones & Bartlett, pp. 257–262, ISBN 978-0-7637-4234-8.