TITLE: $F_\sigma$-subsets in a normal space can be separated
QUESTION [8 upvotes]: Let $X$ be a normal Hausdorff space and let $C,D$ be two $F_{\sigma}$ subsets of $X$
such that $\overline{C} \cap D = \emptyset$ and $C \cap \overline{D} = \emptyset$.
Prove there exists disjoint open subsets $U,V$ such that $C \subset U$ and $D \subset V$.
No idea how to show this. Can you please help?

REPLY [5 votes]: You want to use the same 'climbing a chimney' technique that's used to prove that every regular Lindelöf space is normal. (You might want to stop reading here and take a look at that proof to see whether you can adapt the idea on your own.)
Let $C = \bigcup \limits_{n \in \omega} C_n$ and $D = \bigcup \limits_{n \in \omega} D_n$, where the sets $C_n$ and $D_n$ are closed and for each $n \in \omega$ $C_n \subseteq C_{n+1}$ and $D_n \subseteq D_{n+1}$. By normality there are open sets $V_0$ and $W_0$ such that $C_0 \subseteq V_0 \subseteq \text{cl } V_0 \subseteq X \setminus D$ and $D_0 \subseteq W_0 \subseteq \text{cl } W_0 \subseteq X \setminus C$. Given $V_n$ and $W_n$ for some $n \in \omega$, use normality to get open sets $V_{n+1}$ and $W_{n+1}$ such that 
(1) $C_{n+1} \cup \text{cl } V_n \subseteq V_{n+1} \subseteq \text{cl } V_{n+1} \subseteq X \setminus (\text{cl } D \cup \text{cl } W_n)$
and
(2) $D_{n+1} \cup \text{cl } W_n \subseteq W_{n+1} \subseteq \text{cl } W_{n+1} \subseteq X \setminus (\text{cl } C \cup \text{cl } V_n)$.
Now let $V = \bigcup \limits_{n \in \omega} V_n$ and $W = \bigcup \limits_{n \in \omega} W_n$; these are disjoint open sets containing $C$ and $D$, respectively.