Nagata–Smirnov metrization theorem

The theorem states that a topological space

is called a regular space if every non-empty closed subset

admit non-overlapping open neighborhoods.

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.