 Separation of disjoint compact subsets

Two disjoint compact subsets of a Hausdorff space have disjoint open neighbourhoods. For every $p\in P$, separation of a point and a compact set gives disjoint open sets $G_p\ni p$ and $H_p\supseteq Q$. Choose finitely many $G_p$ covering $P$, then take their union for $U$ and the matching finite intersection of the $H_p$ for $V$. The compactness of both subsets is used, once in the point-versus-compact set result and once in the outer finite selection. The ambient Hausdorff space need not be compact or a normal space.