If $\displaystyle C$ and $\displaystyle D$ are closed subsets in $\displaystyle X$ with $\displaystyle C \cap D = \emptyset$, then $\displaystyle \exists$ open sets $\displaystyle U$ and $\displaystyle V$, s.t $\displaystyle U \supset C$ and $\displaystyle V \supset D$ with $\displaystyle U \cap V = \emptyset$

I am trying to use the whole hypothesis.

Certainly, I can say

$\displaystyle X-D$ is open and $\displaystyle X-D \supset C$

$\displaystyle X-C$ is open and $\displaystyle X-C \supset D$

But these choices of U and V dont work.

$\displaystyle (X-D) \cap (X-C) \neq \emptyset$

What set do I want to consider?

$\displaystyle C \cap D$ and $\displaystyle C \cup D$ are both closed.

Thank you for your help.