Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm searching for some nice applications of Tietze extension theorem, in any area of mathematics. Can you name some (and possibly give references) to me?

Thank you in advance!

share|improve this question
Not really an application, but a nice sharpening is Whitney's extension theorem which I found quite surprising when I saw it for the first time. The original article is very nice and definitely worth having a look at. –  t.b. May 28 '11 at 20:23
The Tietze Extension theorem can be used to show that the adjunction of two normal spaces is normal assuming of course that you glue on a closed subset of the spaces. –  JSchlather May 29 '11 at 1:42

2 Answers 2

One application I particularly like, from an undergraduate analysis exam problem:

Theorem: A metric space $X$ is compact if and only if every continuous real-valued function on $X$ is bounded.

Proof: Assume first $X$ is compact. If $f:X\to \mathbb R$ is continuous and unbounded, then we have some sequence $(x_n)$ in $X$ such that $f(x_n)>n,\forall n\in\mathbb N$. Since $X$ is compact, we have some convergent subsequence $(x_{n_k})$, so $\lim\limits_{k\to\infty}f(x_{n_k})=f(\lim\limits_{k\to\infty}x_{n_k})$. But this is impossible, as $f(x_{n_k})\to\infty$, hence any continuous real-valued function is bounded. If instead $X$ is not compact, then we have some sequence $(x_n)$ in $X$ which has no convergent subsequence. Hence every convergent sequence with terms in the set $S=\{x_1,x_2,\ldots\}$ must be eventually constant, so has limit in $S$, hence $S$ is closed. Define the function $f:S\to \mathbb R$ by $f(x_n)=n$, which is continuous because $S$ is a discrete set. By the Tietze extension theorem, we can extend $f$ to a continuous unbounded function $g:X\to\mathbb R$.

share|improve this answer

I'm not sure if the following proof works, but it uses the Tietze extension theorem, among other things, to prove the density of $C_{0}(\mathbb{R})$ in $L^{1}$.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.