# Metrizable iff Regular and has Sigma-Locally Finite Basis

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## Theorem

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

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

## Proof

## Sources

- 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