# Nagata-Smirnov Metrization Theorem

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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 basis that is countably locally finite.

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Source of Name

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

## Sources

- 1975: James R. Munkres:
*Topology*