Compactness and open sets

Question

I have this small question, if $(E,\tau)$ is a Hausdorff space and $A,B$ two separated compact sets, how to prove the existence of two open disjoint sets $U$ and $V$ such that $B\subset V$ and $A\subset U$ ?

Please

Thank you.

Answer 1

For each pair of points $p\in A$ and $q\in B$, since the space is Hausdorff, there are open sets $U_{pq}$ and $V_{pq}$ such that $p\in U_{pq}$ and $q\in V_{pq}$ and the intersection of $U_{pq}$ and $V_{pq}$ is empty.

Fix $p\in A$, then consider $V_{pq}$ as $q$ varies over the elements of $B$. This forms an open cover of $B$ so as $B$ is compact, there is a finite subcover. Let $V_p$ be the union of the sets $V_{pq}$ in this subcover. Now, let $U_p$ be the intersection of the $U_{pq}$ for the $q$'s that appear in this subcover. Then you have open sets $U_p$ and $V_p$ such that $V_p$ contains $B$ and $U_p$ contains $p$ and $V_p$ and $U_p$ are disjoint.

Now, use a similar argument to complete the proof.