= Solution
Let $F,G$ be disjoint closed <subsets> of $K$. They are <compact>. For each $x\in F,y\in G$, the <Hausdorff space> condition gives disjoint open neighbourhoods $U_{xy},V_{xy}$. For fixed $x$, finitely many $V_{xy}$ cover $G$; intersect the corresponding $U_{xy}$ and take the <union> of those $V_{xy}$ to obtain disjoint open <sets> $U_x\ni x$ and $V_x\supset G$. Finitely many $U_x$ cover $F$. Their <union> and the <intersection> of the corresponding $V_x$ are disjoint open neighbourhoods of $F,G$. The case of an empty <set> is immediate. Hence \b[$K$ is a <normal topological space>].
<Urysohn lemma> states that in a <normal topological space> satisfying $T_1$, disjoint closed <sets> $F,G$ admit a <continuous function> $h:K\to[0,1]$ with $h=0$ on $F$ and $h=1$ on $G$.
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