Exercise 16, § 2

Let $H$ be a subgroup of a semi-topological group $G$, contained in the component $K$ of $e$. Show that the components of the space $G/H$ are the images of the components of $G$ under the canonical mapping $f$ of $G$ onto $G/H$ [if $L$ is a component of $G/H$, show that $\overline{f}(L)$ is connected, by reductio ad absurdum]. Show that $K$ is the smallest of the closed subgroups $H$ of $G$ such that $G/H$ is totally disconnected.