In
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, the Nagata–Smirnov metrization theorem characterizes when a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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, -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 -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 and 1951,
[Y. Smirnov, "A necessary and sufficient condition for metrizability of a topological space" (Russian), ''Dokl. Akad. Nauk SSSR'' 77 (1951), 197–200.] respectively.
See also
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Notes
References
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{{DEFAULTSORT:Nagata-Smirnov metrization theorem
General topology
Theorems in topology