= Separation of disjoint compact subsets
{title2=$P\cap Q=\varnothing\ \Longrightarrow\ P\subseteq U,\ Q\subseteq V,\ U\cap V=\varnothing$}
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>.
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