# Nagata-Smirnov Metrization Theorem

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The **Nagata-Smirnov Metrization Theorem** gives a full characterization of metrizable topological spaces.^{[1]} Essentially, the Nagata-Smirnov Metrization Theorem states that the regularity of a space $X$ and the existence of a countably locally finite basis for $X$ are equivalent to metrizability of $X$.

## Definition

A topological space $X$ is metrizable if and only if $X$ is regular and has a basis that is countably locally finite.^{[2]}

## Further Reading

## See Also

## External Links

## References

- ↑ Leeb, William. July, 2007. The Nagata-Smirnov Metriztion Theorem - University of Chicago Mathematics
- ↑ Munkres, James R. Topology, 2015. pg. 248-249.