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
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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