Bing metrization theorem

The Bing metrization theorem in topology characterizes when a topological space is metrisable. The theorem states that a topological space $X$ is metrisable 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 $F$ of subsets of a space $X$ is called discrete, when every point of $X$ has a neighbourhood that intersects at most finite many subsets from $F$.

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.

The theorem was proven by Bing in 1951 and was an independent discovery with the Nagata-Smirnov metrisation theorem that was proved independently by both Nagata (1950) and Smirnov (1951). Both theorems are often merged in the Bing-Nagata-Smirnov metrisation theorem. It is a common tool to prove other metrisation theorems, e.g. the Moore metrisation theorem: a collectionwise normal, Moore space is metrisable, is a direct consequence.

References

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