The Urysohn Metrization Theorem gives conditions under which a topological space can be given a metric that induces its topology, stating that every regular, second-countable topological space is metrizable. Named for Pavel Urysohn, it is a foundational result of general topology.
Facts
StatementEvery regular, second-countable Hausdorff topological space is metrizable, the commonly cited modern form of Urysohn's metrization theorem. Urysohn's own 1925 posthumous paper proved the narrower case of every second-countable normal Hausdorff space; the regular-space form given here was completed by Tikhonov in 1926. 1 Connections
Sources
1. Urysohn Metrization Theorem (Wikipedia)
Wikimedia FoundationMetrization theorems section, historical note
The form of the theorem shown here was in fact proved by Tikhonov in 1926. What Urysohn had shown, in a paper published posthumously in 1925, was that every second-countable normal Hausdorff space is metrizable.
Metrization theorems section, historical note, second sentence
What Urysohn had shown, in a paper published posthumously in 1925, was that every second-countable normal Hausdorff space is metrizable.
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