# Difference between revisions of "Urysohn Lemma"

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− | The <strong>Urysohn Lemma</strong> asserts the existence of certain real-valued [[Continuous|continuous]] [[Function|function]]s on a [[Normal Space|normal space]] $X$. The Urysohn Lemma has several significant implications, among them the [[ | + | The <strong>Urysohn Lemma</strong> asserts the existence of certain real-valued [[Continuous|continuous]] [[Function|function]]s on a [[Normal Space|normal space]] $X$. The Urysohn Lemma has several significant implications, among them the [[Tietze Extension Theorem]] and the [[Urysohn Metrization Theorem]]. |

== Formal Statement == | == Formal Statement == | ||

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* [[Normal Space|Normal Space]]s | * [[Normal Space|Normal Space]]s | ||

* [[Completely Normal Space|Completely Normal Space]]s | * [[Completely Normal Space|Completely Normal Space]]s | ||

− | * [[ | + | * [[Tietze Extension Theorem]] |

== See Also == | == See Also == |

## Latest revision as of 06:26, 1 December 2018

The **Urysohn Lemma** asserts the existence of certain real-valued continuous functions on a normal space $X$. The Urysohn Lemma has several significant implications, among them the Tietze Extension Theorem and the Urysohn Metrization Theorem.

## Formal Statement

Let $X$ be a normal space and let $A$ and $B$ be disjoint closed subsets of $X$. Let $[a,b]$ be a closed interval in the real line. Then there exists a continuous map $$f : X \to [a,b]$$ such that $f(x) = a$ for every $x$ in $A$, and $f(x) = b$ for every $x$ in $B$.^{[1]}

## Further Reading

## See Also

## References

- ↑ Munkres, James R. Topology, 2015. pg. 205.