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 ${\displaystyle X}$ is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (that is, 𝜎-locally finite) basis.

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

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[1] and 1951,[2] 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

1. J. Nagata, "On a necessary and sufficient condition of metrizability", J. Inst. Polytech. Osaka City Univ. Ser. A. 1 (1950), 93–100.
2. Y. Smirnov, "A necessary and sufficient condition for metrizability of a topological space" (Russian), Dokl. Akad. Nauk SSSR 77 (1951), 197–200.

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.