In topology, the Bing metrization theorem, named after R. H. Bing, characterizes when a topological space is metrizable.
Contents
Formal statement
The theorem states that a topological space
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.