# Difference between revisions of "Nagata-Smirnov Metrization Theorem"

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− | + | The [[Nagata-Smirnov Metrization Theorem]] gives a full characterization of [[Metrizable|metrizable]] [[Topological space|topological space]]s.<ref>Leeb, William. July, 2007. [http://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALAPP/Leeb.pdf The Nagata-Smirnov Metriztion Theorem - University of Chicago Mathematics]</ref> Essentially, the Nagata-Smirnov Metrization Theorem states that the [[Regular Space|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|topological space]] $X$ is [[Metrizable|metrizable]] if and only if $X$ is [[Regular Space|regular]] and has a [[Basis|basis]] that is [[Countably Locally Finite Basis|countably locally finite]].<ref>Munkres, James R. Topology, 2015. pg. 248-249.</ref> | ||

+ | |||

+ | == Further Reading == | ||

+ | * [[Metric space|Metric Space]]s | ||

+ | |||

+ | == See Also == | ||

+ | * [[Urysohn Lemma]] | ||

+ | * [[Urysohn Metrization Theorem]] | ||

+ | * [[Tychonoff Theorem]] | ||

+ | |||

+ | == External Links == | ||

+ | * [https://arxiv.org/abs/1311.4940 A Note on the Metrizability of Spaces - Weiss, Ittay (ArXiv)] | ||

+ | |||

+ | == References == | ||

+ | <references group="references" /> | ||

[[Category:General Topology]] | [[Category:General Topology]] |

## Latest revision as of 06:48, 1 December 2018

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.