# Difference between revisions of "Urysohn Metrization Theorem"

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− | + | If a [[Topological space|topological space]] $X$ satisfies a certain countability axiom (the second) and a certain [[Separation Axioms|separation axiom]] (the regularity axiom), then $X$ can be imbedded in a [[Metric space|metric space]] and is thus ''metrizable''.<ref>Munkres, James R. Topology, 2015. pg. 187.</ref> | |

+ | |||

+ | == Further Reading == | ||

+ | * [[Metric space|Metric Spaces]] | ||

+ | * [[Separation Axioms]] | ||

+ | * [[Topological space|Topological Spaces]] | ||

+ | |||

+ | == External Links == | ||

+ | * [https://ncatlab.org/nlab/show/Urysohn+metrization+theorem nLab - Urysohn Metrization Theorem] | ||

+ | * [http://www.math.toronto.edu/ivan/mat327/docs/notes/15-umt.pdf Urysohn's Metrization Theorem - University of Toronto Mathematics (pdf)] | ||

+ | * [https://math.arizona.edu/~gradprogram/workshops/integration/projects/urysohn.pdf University of Arizona Mathematics (pdf)] | ||

+ | |||

+ | == References == | ||

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

+ | |||

+ | [[Category:General Topology]] |

## Latest revision as of 05:05, 1 December 2018

If a topological space $X$ satisfies a certain countability axiom (the second) and a certain separation axiom (the regularity axiom), then $X$ can be imbedded in a metric space and is thus *metrizable*.^{[1]}

## Further Reading

## External Links

- nLab - Urysohn Metrization Theorem
- Urysohn's Metrization Theorem - University of Toronto Mathematics (pdf)
- University of Arizona Mathematics (pdf)

## References

- ↑ Munkres, James R. Topology, 2015. pg. 187.