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{{Short description|Characterizes when a topological space is metrizable}}
In [[topology]], the '''Bing metrization theorem''', named after [[R. H. Bing]], characterizes when a [[topological space]] is [[metrizable]].

== Formal statement ==

The theorem states that a topological space <math>X</math> is metrizable if and only if it is [[Regular space|regular]] and '''[[Kolmogorov axiom|T<sub>0</sub>]]''' and has a σ-discrete [[Basis (topology)|basis]]. A family of sets is called σ-discrete when it is a union of countably many discrete collections, where a family <math>\mathcal{F}</math> of subsets of a space <math>X</math> is called discrete, when every point of <math>X</math> has a neighborhood that intersects at most one member of <math>\mathcal{F}.</math> 

== History ==

The theorem was proven by [[RH Bing|Bing]] in 1951 and was an independent discovery with the [[Nagata–Smirnov metrization theorem]] that was proved independently by both [[Jun-iti Nagata|Nagata]] (1950) and Smirnov (1951). Both theorems are often merged in the Bing-Nagata-Smirnov metrization theorem. It is a common tool to prove other [[metrization theorem]]s, e.g. the Moore metrization theorem – a [[collectionwise normal]], [[Moore space (topology)|Moore space]] is metrizable – is a direct consequence.

== Comparison with other metrization theorems ==

Unlike the Urysohn's [[metrization theorem]] which provides a sufficient condition for metrization, this theorem provides both a necessary and sufficient condition for a [[topological space]] to be [[metrizable]].

== See also ==

* {{annotated link|Metrizable topological vector space}}
* {{annotated link|Nagata–Smirnov metrization theorem}}

== References ==

* "General Topology", Ryszard Engelking, Heldermann Verlag Berlin, 1989. {{ISBN|3-88538-006-4}}

[[Category:Theorems in topology]]

[[de:Satz von Bing-Nagata-Smirnow]]