Nagata–Smirnov metrization theorem explained

In topology, the Nagata–Smirnov metrization theorem characterizes when a topological space is metrizable. The theorem states that a topological space

X is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (that is, -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 (or -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 (independent) proofs were published in 1950^[1] and 1951,^[2] respectively.

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Notes and References

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.