6.3Metrization Theorems

Updated 23 Aug 2026Chapter (PDF)

We only cover local finiteness and paracompactness to the extent necessary to generalize the mechanism of partitions of unity to spaces with a condition weaker than compactness. We also briefly state the major metrization theorems (Nagata-Smirnov and Smirnov) and their immediate consequences for manifolds, while blackboxing the highly technical proofs.

The Urysohn metrization theorem gave us a sufficient condition for metrizability (regular and second-countable), but mathematicians are rarely satisfied until conditions are both necessary and sufficient. The following theorems, whose proofs we omit as they are largely technical extensions of Urysohn's methods, provide exact characterizations of metrizable spaces.

[6.3.1]Theorem(Nagata-Smirnov Metrization Theorem)#

A space $X$ is metrizable if and only if $X$ is regular and admits a countably locally finite basis.

[6.3.2]Definition(Locally Metrizable)#

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

[6.3.3]Theorem(Smirnov Metrization Theorem)#

A space $X$ is metrizable if and only if it is a paracompact Hausdorff space that is locally metrizable.

[6.3.4]Remark(Manifolds are Metrizable)#

Smirnov's theorem is particularly elegant when applied to manifolds. Because an $m$ -manifold is locally homeomorphic to $\bR^m$ , it is automatically locally metrizable. Thus, a manifold is globally metrizable if and only if it is paracompact Hausdorff. Since standard definitions of manifolds require them to be second-countable (which implies they are Lindelof, and regular Lindelof spaces are paracompact), manifolds are automatically metrizable!