4.6Tietze Extension Theorem

Updated 21 Aug 2026Chapter (PDF)

Another useful consequence of the Urysohn lemma is the useful theorem called the Tietze extension theorem. A common problem in analysis is being able to extend a construction to a larger space, while still keeping important properties intact. For a space $X$ , we may have a continuous, real-valued function $f: A \to \bR$ , with $A$ a subset of $X$ . We ask if there is a way to extend $f$ to some function $\tilde{f}: X \to \bR$ , such that $\tilde{f}|_A = f$ and that continuity is preserved. Of course, this is not true for all spaces $X$ , but certain restrictions on $X$ and $A$ make this theorem true. This is what the Tietze extension theorem is concerned with.

[4.6.1]Theorem(Tietze Extension Theorem)#

Suppose $X$ is a normal space and $A \subseteq X$ is closed. Then any continuous map $f: A \to [a, b]$ may be extended to a continuous map $f: X \to [a, b]$ . The result holds if the closed interval in the codomain is replaced with $\bR$ .

The proof will be omitted for now.