In topology, the Bing metrization theorem, named after R. H. Bing, characterizes when a topological space is metrizable.
The theorem states that a topological space
X
l{F}
X
X
l{F}.
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.
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.