In topology, the Bing metrization theorem, named after R. H. Bing, characterizes when a topological space is metrizable.
Formal statement
editThe theorem states that a topological space is metrizable if and only if it is regular and T0 and has a
History
editThe 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
editUnlike 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
edit- Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
- Nagata–Smirnov metrization theorem – Characterizes when a topological space is metrizable
References
edit- "General Topology", Ryszard Engelking, Heldermann Verlag Berlin, 1989. ISBN 3-88538-006-4