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.

Definition: 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.