Nagata–Smirnov metrization theorem
Characterizes when a topological space is metrizable / From Wikipedia, the free encyclopedia
In topology, the Nagata–Smirnov metrization theorem characterizes when a topological space is metrizable. The theorem states that a topological space 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 is called a regular space if every non-empty closed subset
of
and a point p not contained in
admit non-overlapping open neighborhoods.
A collection in a space
is countably locally finite (or 𝜎-locally finite) if it is the union of a countable family of locally finite collections of subsets of
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.