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# Tietze extension theorem

Let $X$ be a topological space. Then the following are equivalent:

1. $X$ is normal.

2. If $A$ is a closed subset in $X$, and $f\colon A\to[-1,1]$ is a continuous function, then $f$ has a continuous extension to all of $X$. (In other words, there is a continuous function $f^{\ast}\colon X\to[-1,1]$ such that $f$ and $f^{\ast}$ coincide on $A$.)

*Remark:*
If $X$ and $A$ are as above, and $f\colon A\to(-1,1)$ is a continuous function, then $f$ has a continuous extension to all of $X$.

The present result can be found in [1].

# References

- 1
A. Mukherjea, K. Pothoven,
*Real and Functional analysis*, Plenum press, 1978.

Related:

ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

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