Exercise 3, § 9

a) Let $X$ be a Hausdorff space, and let $A, B$ be two disjoint compact subsets of $X$. Show that there is a neighbourhood of $A$ and a neighbourhood of $B$ which have no common point.

b) Let $X$ be a regular space and $A$ a compact subset of $X$, and let $B$ be a closed subset of $X$ which does not meet $A$. Show that there is a neighbourhood of $A$ and a neighbourhood of $B$ which have no common point.