Exercise 8, § 3

Let $G$ be a topological group and let $H$ be a closed subgroup of $G$, contained in the centre of $G$, and such that there is a continuous surjective homomorphism $\varphi : \mathbf{R} \to G/H$. Show that $G$ is abelian [if $a_0$ is an element of $G$ which is not in $H$, observe that there exists a sequence $(a_n)$ of elements of $G$ and a sequence $(c_n)$ of elements of $H$ such that $a_n = c_n a_{n+1}$ and that the subgroup of $G$ generated by $H$ and the $a_n$ is dense in $G$].