Urysohn metrization theorem

This article is about a metrization theorem: a theorem that gives necessary and sufficient conditions for a metric (possibly with additional restrictions) to exist. In particular, it gives some conditions under which a topological space is metrizable.

Statement

Any regular second-countable topological space is metrizable.

Related results

• Nagata-Smirnov metrization theorem is a more powerful statement, which weakens second-countability to having a countable locally finite basis, and shows that the new set of conditions is both necessary and sufficient.