Exercise 18, § 6

Let $K$ be a Hausdorff topological division ring, let $A$ be a closed subset of $K$ and let $B$ be a compact subset of $K$ such that $o \notin B$. Show that $AB$ and $BA$ are closed in $K$ (cf. § 4, no. 1, Proposition 1, Corollary 1). \* In the field $\mathbf{R}$ of real numbers, give an example where $A$ is closed, and $B$ is compact, $o \in B$ and $AB$ is not closed. \*