Bing metrization theorem

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 $X$ is metrizable if and only if it is regular and T₀ and has a σ-discrete basis. A family of sets is called σ-discrete when it is a union of countably many discrete collections, where a family ${\mathcal {F}}$ of subsets of a space $X$ is called discrete, when every point of $X$ has a neighborhood that intersects at most one member of ${\mathcal {F}}$ .

History

The theorem was proven by Bing in 1951 and was an independent discovery with the Nagata-Smirnov metrization theorem that was proved independently by both 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 theorems, e.g. the Moore metrization theorem: a collectionwise normal, 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.

References

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

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