Difference between revisions of "Tietze Extension Theorem"

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The <strong>Tietze Extension Theorem</strong> deals with the problem of extending a [[Continuous|continuous]] real-valued [[Function|function]] that is defined on a [[Subspace|subspace]] of a [[Topological space|topological space]] $X$ to a continuous function defined on all of $X$.
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__NOTOC__
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==Theorem==
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Let $(X,\tau)$ be a [[normal]] [[topological space]], let $A \subset X$ be [[closed]], and let $f \colon A \rightarrow \mathbb{R}$ be a [[continuous]] [[function]] (where $\mathbb{R}$ is equipped with the [[Euclidean topology]]). Then there exists a continuous function $F \colon X \rightarrow \mathbb{R}$ such that for all $a \in A$, $F(a)=f(a)$.
  
== Definition ==
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==Proof==
Let $X$ be a [[Normal Space|normal space]] and let $A$ be a [[Closed|closed]] [[Subspace|subspace]] of $X$.
 
# Any [[Continuous Map|continuous map]] of $A$ into the [[Closed Interval|closed interval]] $[a,b]$ of $\mathbb{R}$ may be extended to a continuous map of all of $X$ onto $[a,b]$.
 
# Any continuous map of $A$ into $\mathbb{R}$ may be extended to a continuous map of all of $X$ into $\mathbb{R}$.
 
  
== Further Reading ==
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== See Also ==
 
* [[Urysohn Lemma]]
 
* [[Urysohn Lemma]]
 
== See Also ==
 
 
* [[Urysohn Metrization Theorem]]
 
* [[Urysohn Metrization Theorem]]
  
 
== External Links ==
 
== External Links ==
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* [https://ncatlab.org/nlab/show/Tietze+extension+theorem nLab - Tietze Metrization Theorem]
  
 
== References ==
 
== References ==
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[[Category:General Topology]]
 
[[Category:General Topology]]
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[[Category:Theorem]]
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[[Category:Unproven]]

Latest revision as of 02:03, 24 December 2018

Theorem

Let $(X,\tau)$ be a normal topological space, let $A \subset X$ be closed, and let $f \colon A \rightarrow \mathbb{R}$ be a continuous function (where $\mathbb{R}$ is equipped with the Euclidean topology). Then there exists a continuous function $F \colon X \rightarrow \mathbb{R}$ such that for all $a \in A$, $F(a)=f(a)$.

Proof

See Also

External Links

References