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 [[Tieze Extension Theorem]] and the [[Urysohn Metrization Theorem]].
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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 [[Tietze Extension Theorem]] and the [[Urysohn Metrization Theorem]].
  
 
== Formal Statement ==
 
== Formal Statement ==
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== Further Reading ==
 
== Further Reading ==
 
* [[Normal Space|Normal Space]]s
 
* [[Normal Space|Normal Space]]s
* [[Tieze Extension Theorem]]
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* [[Completely Normal Space|Completely Normal Space]]s
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* [[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

  1. Munkres, James R. Topology, 2015. pg. 205.