Is it always possible to extend continuous functions defined on a closed subset of a locally compact Hausdorff space?

Question

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).

You cannot in general extend all functions on closed sets: the existence of non-normal LCH spaces shows this. — Henno Brandsma, Feb 18 '22 at 05:27
@HennoBrandsma Yes. But then how did the authors use Tietze in the proof?? — sigma, Feb 18 '22 at 05:34
they must use some weaker version (a positive extension). Look in the index to find the exact theorem they’re quoting. — Henno Brandsma, Feb 18 '22 at 05:37
@HennoBrandsma the Lemma is in the paper "Sub-Stonean Spaces and Corona Sets" by K Grove and Gert Pedersen (Lemma 1.3).. — sigma, Feb 18 '22 at 05:40
I don’t have nor know that paper. Cannot you quote the lemma in the post to be self-contained? — Henno Brandsma, Feb 18 '22 at 05:45

score 6 · Accepted Answer · answered Feb 18 '22 at 05:50

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 $1$.)

I missed the $C_0(X)$ part... good spot – Henno Brandsma Feb 18 '22 at 10:52 — Henno Brandsma, Feb 18 '22 at 10:52

Is it always possible to extend continuous functions defined on a *closed* subset of a locally compact Hausdorff space?

1 Answers1

Is it always possible to extend continuous functions defined on a closed subset of a locally compact Hausdorff space?