Applications of Tietze Extension Theorem 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!
 A: 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 $L^{1}$.
A: 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 $g:X\to\mathbb R$.
