Exercise on compact $G_\delta$ sets

Question

I'm having trouble proving an exercise in Folland's book on real analysis.

Problem: Consider a locally compact Hausdorff space $X$. If $K\subset X$ is a compact $G_\delta$ set, then show there exists a $f\in C_c(X, [0,1])$ with $K=f^{-1}(\{1\})$.

We can write $K=\cap_1^\infty U_i$, where the $U_i$ are open.

My thought was to use Urysohn's lemma to find functions $f_i$ which are 1 on $K$ and $0$ outside of $U_i$, but I don't see how to use them to get the desired function. If we take the limit, I think we just get the characteristic function of $K$.

I apologize if this is something simple. It has been a while since I've done point-set topology.

score 3 · Accepted Answer · answered Jul 07 '12 at 02:57

3

As you have said, we can use Urysohn's lemma for compact sets to construct a sequence of functions $f_i$ such that $f_i$ equals $1$ in $K$ and $0$ outside $U_i$.

Furthermore, $X$ is locally compact, so there is an open neighbourhood $U$ of $K$ whose closure is compact. We can then assume without loss of generality that $U_i\subseteq U$

Then we can put $f=\sum_i2^{-i} f_i$. Clearly, $f^{-1}[\{1\}]=K$. Moreover, $f$ is the uniform limit of continuous functions (because $f_i$ are bounded by $1$), so it is continuous, and its support is contained in $U$, so $f$ is the function you seek.

answered Jul 07 '12 at 02:57

tomasz

35,474

Excellent, thank you. – Potato Jul 07 '12 at 02:58
How can you assume that $U_i \subseteq U\ $? – Anacardium Dec 31 '20 at 08:44
1

@Anacardium: because $K\subseteq U$, so $K=U\cap K=U\cap \bigcap_i U_i=\bigcap_i(U\cap U_i)$. – tomasz Dec 31 '20 at 16:24
Another question. We know that Urysohn's lemma is only valid for normal topological space and the locally compact Hausdorff space is not necessarily normal. Then how can we apply Urysohn's lemma here? – Anacardium Dec 31 '20 at 17:23
@Anacardium $\bar U$ is compact. – tomasz Dec 31 '20 at 17:39
But then you can only define $f$ on $\overline {U}.$ Not on the whole of $[0,1].$ I can't get your point. – Anacardium Dec 31 '20 at 18:38
@Anacardium: you can just put $0$ outside of $\bar U$. – tomasz Dec 31 '20 at 18:48

Exercise on compact $G_\delta$ sets

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