View source for Nagata–Smirnov metrization theorem

{{Short description|Characterizes when a topological space is metrizable}}
In [[topology]], the '''Nagata–Smirnov metrization theorem''' characterizes when a [[topological space]] is [[metrizable]].  The theorem states that a topological space <math>X</math> is metrizable if and only if it is [[Regular space|regular]], [[Hausdorff_space|Hausdorff]] and has a [[countably locally finite]] (that is, {{sigma}}-locally finite) [[Basis (topology)|basis]].

A topological space <math>X</math> is called a regular space if every non-empty closed subset <math>C</math> of <math>X</math> and a point p not contained in <math>C</math> admit non-overlapping open neighborhoods.
A collection in a space <math>X</math> is countably locally finite (or {{sigma}}-locally finite) if it is the union of a countable family of locally finite collections of subsets of <math>X.</math> 

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 [[Jun-iti Nagata|Junichi Nagata]] and [[Yuri Mikhailovich Smirnov|Yuriĭ Mikhaĭlovich Smirnov]], whose (independent) proofs were published in 1950<ref>J. Nagata, [https://core.ac.uk/download/pdf/35260599.pdf "On a necessary and sufficient condition of metrizability"], ''J. Inst. Polytech.'' Osaka City Univ. Ser. A. '''1''' (1950), 93–100.</ref> and 1951,<ref>Y. Smirnov, "A necessary and sufficient condition for metrizability of a topological space" (Russian), ''Dokl. Akad. Nauk SSSR'' '''77''' (1951), 197–200.</ref> respectively.

== See also ==

* {{annotated link|Bing metrization theorem}}
* {{annotated link|Kolmogorov's normability criterion}}
* {{annotated link|Uniformizable space}}

== Notes ==

{{reflist}}

==References==

* {{citation|title=Topology|first=James R.|last=Munkres|authorlink=James Munkres|publisher=Prentice Hall|year=1975|isbn=0-13-925495-1|contribution=Sections 6-2 and 6-3|pages=[https://archive.org/details/topologyfirstcou00munk_0/page/247 247–253]|url-access=registration|url=https://archive.org/details/topologyfirstcou00munk_0/page/247}}.
* {{citation|title=Foundations of Topology|first=C. Wayne|last=Patty|edition=2nd|publisher=Jones & Bartlett|year=2009|isbn=978-0-7637-4234-8|contribution=7.3 The Nagata–Smirnov Metrization Theorem|pages=257–262}}.

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{{DEFAULTSORT:Nagata-Smirnov metrization theorem}}
[[Category:General topology]]
[[Category:Theorems in topology]]