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 $X$.
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== Definition ==
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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>
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== Further Reading ==
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* [[Metric space|Metric Space]]s
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== See Also ==
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* [[Urysohn Lemma]]
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* [[Urysohn Metrization Theorem]]
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* [[Tychonoff Theorem]]
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== External Links ==
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* [https://arxiv.org/abs/1311.4940 A Note on the Metrizability of Spaces - Weiss, Ittay (ArXiv)]
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== References ==
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<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

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