If $X$ is a Hausdorff Space and there are 2 disjoint compact sets $A,B\subset X$, we want to prove $\exists V,U \subset X, V \bigcap U=\emptyset $ S.T. $A\subset U, B\subset V$.($U,V$ are open sets)
My idea is as follows:
We choose arbitray 2 points, $x_1\in A, y_1\in B$ and by definition of compactness, we can find a neighborhood of $x_1$ and $y_1$, denoted as $U_1,V_1$ S.T. $U_1\bigcap U_2=\emptyset$. By the same algorithm, we will have $U=\{U_1, U_2,...\},V=\{V_1,V_2,...\}$. So, what I want to ask is that are $U,V$ still disjoint? If it is then problem solved.
It might make it a little easier to prove this in two pieces.
First show that you can separate any point $x\in A$ from the entire set $B$ with disjoint open sets.
Explicitly, show that there always exists $\mathcal{O}_x,\mathcal U_x$ disjoint and open such that $x\in \mathcal O_x$ and $B\subseteq \mathcal U_x$.
Then use the pairs of open sets generated this way to separate $A$ as a whole from $B$.