Nagata-Smirnov Metrization Theorem

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Then:

$T$ is metrizable if and only if $T$ is regular and has a basis that is $\sigma$-locally finite.

Proof

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Source of Name

This entry was named for Jun-iti Nagata and Yurii Mikhailovich Smirnov.

Sources

• 1955: John L. Kelley: General Topology: Chapter $4$: Embedding and Metrization
• 1970: Stephen Willard: General Topology: Chapter $7$: Metrizable Spaces: $\S23$: Metrization: Theorem $23.9$
• 1975: James R. Munkres: Topology: Chapter $6$: Metrization Theorems and Paracompactness: $\S40$: The Nagata-Smirnov Metrization Theorem: Theorem: $40.3$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metrizability