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