In the following lemma the authors used Tietze's extension to get $f_1$ and $g_1$.
I know this version of Tietze, but it requires the subset to be compact not merely closed, i.e., continuous functions defined on compact subsets of a LCH space can be extended upto the whole space. But in the above Lemma 1.3 the subset $X_0$ was only assumed to be closed; then how did the authors use the Tietze theorem?
Question. Suppose $X_0$ is a closed subset of a locally compact Hausdorff space $X$, and $f\in C_0(X_0)$. Is it always possible to extend $f$ to whole of $X$?
EDIT The lemma above is in this pdf (Lemma 1.3).
Best Answer
The key extra assumption here is that you are extending not just a continuous function but a continuous function that vanishes at infinity. If $f\in C_0(X_0)$, then $f$ can be thought of as a function on the one-point compactification $X_0\cup\{\infty\}$ with $f(\infty)=0$. Now $X_0\cup\{\infty\}$ embeds in the one-point compactification $X\cup\{\infty\}$ (as just the subspace $X_0$ plus the point at infinity) as a closed subspace, so you can extend $f$ to all of $X\cup\{\infty\}$ using the usual Tietze extension theorem.
(Without this extra assumption the result is not true. For instance, if $X$ is any locally compact Hausdorff space that is not normal and $A,B\subseteq X$ are disjoint closed subsets that cannot be separated by open sets, then you could take $X_0=A\cup B$ and $f$ that sends $A$ to $0$ and $B$ to $1$.)