Exercise 20, § 11

Let $X$ be a Hausdorff space in which each point has a fundamental system of neighbourhoods which are both open and closed in $X$. Let $Y$ be a subspace of $X$, and let $A$ be a compact subset of $Y$ which is both open and closed in $Y$; show that there is a subset $B$ of $X$ which is both open and closed in $X$, such that $B \cap Y = A$.