# 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 $\mathbb{R}$. | + | # 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 == | ||

+ | * [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 $X$.

- 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 $[a,b]$.
- 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

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