Difference between revisions of "Tietze Extension Theorem"

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Let $X$ be a [[Normal Space|normal space]] and let $A$ be a [[Closed|closed]] [[Subspace|subspace]] of $X$.
 
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|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 $\mathbb{R}$.
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# Any continuous map of $A$ into $\mathbb{R}$ may be extended to a continuous map of all of $X$ into $\mathbb{R}$.<ref>Munkres, James R. Topology, 2015. pg. 217-220.</ref>
  
 
== Further Reading ==
 
== Further Reading ==
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== External Links ==
 
== External Links ==
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* [https://ncatlab.org/nlab/show/Tietze+extension+theorem nLab - Tietze Metrization Theorem]
  
 
== References ==
 
== References ==

Latest revision as of 06:30, 1 December 2018

The Tietze Extension Theorem deals with the problem of extending a continuous real-valued function that is defined on a subspace of a topological space $X$ to a continuous function defined on all of $X$.

Definition

Let $X$ be a normal space and let $A$ be a closed subspace of $X$.

  1. Any continuous map of $A$ into the closed interval $[a,b]$ of $\mathbb{R}$ may be extended to a continuous map of all of $X$ onto $[a,b]$.
  2. Any continuous map of $A$ into $\mathbb{R}$ may be extended to a continuous map of all of $X$ into $\mathbb{R}$.[1]

Further Reading

See Also

External Links

References

  1. Munkres, James R. Topology, 2015. pg. 217-220.