Here's a question I've been working on:
Suppose that $X$ is a normal topological space, that $F\subseteq X$ is closed, and that $F\subseteq U_1 \cup U_2$ for open sets $U_1,U_2$. Prove that there exist closed sets $F_1,F_2$ with $F=F_1 \cup F_2$, $F_1 \subseteq U_1$, and $F_2 \subseteq U_2$.
I wish I could put up a partial solution, but I don't have much. I'm having a hard time seeing where I could apply normality---where are my disjoint closed sets?
Here's what I do know:
(1) closed subspaces of normal spaces are normal;
(2) $X$ is normal iff given a closed set $F$ and an open set $U$ containing $F$, there exists an open set $V$ such that $F \subseteq V\subseteq \bar{V} \subseteq U$.
I'm thinking that the characterization of normality given in (2) might be more helpful than the actual definition of normality (since it gives me closed sets contained in open sets), but again, I'm having a difficult time applying it.
Any hints would be appreciated.