Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 19F / iii / 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 $K$ 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 $G$.