# Smirnov metrization theorem

The Smirnov metrization theorem establishes necessary and sufficient conditions for a topological space to be metrizable. The theorem reduces questions of metrizability to paracompactness and a local metrizability condition.

A space $X$ is locally metrizable if every point $x\in X$ has a neighborhood that is metrizable in the subspace topology.

Theorem (Smirnov metrization theorem): A space is metrizable if and only if it is paracompact and locally metrizable.

Title Smirnov metrization theorem SmirnovMetrizationTheorem 2013-03-22 18:01:00 2013-03-22 18:01:00 rm50 (10146) rm50 (10146) 7 rm50 (10146) Theorem msc 54E35 locally metrizable