# Metrizable iff Regular and has Sigma-Locally Finite Basis

## Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

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

## Proof

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 5$: Metrizability