Exercise 4, § 2

a) Let $G$ be a locally compact abelian group and let $H$ be a subgroup of $G$ isomorphic to $\mathbf{R}^p \times \mathbf{T}^q$ where $p, q$ are integers $\geqslant 0$; suppose also that $G/H$ is isomorphic to either $\mathbf{R}$ or $\mathbf{T}$. Show that $G$ is isomorphic to the product of $H$ and a closed subgroup $L$ isomorphic to $G/H$. [Apply Chapter V, § 3, Exercise 7 d), to obtain a continuous homomorphism $f$ of $\mathbf{R}$ into $G$ such that $f(\mathbf{R}) \not\subset H$; hence construct another continuous homomorphism $g : \mathbf{R} \to G$ such that $f(t) - g(t) \in H$ for all $t \in \mathbf{R}$ and $g(\mathbf{R}) \cap H = \{ e \}$. Note that $f(H)$ is a proper closed subgroup of $\mathbf{R}$ and that, for every element $z \neq e$ in $H$, there exists a continuous homomorphism $u$ of $\mathbf{R}$ into $H$ such that $u(1) = z .$]

b) Let $G$ be a Hausdorff abelian topological group, let $H$ be a closed subgroup of $G$ isomorphic to $\mathbf{R}^p \times \mathbf{T}^q$, and suppose that there exists a surjective continuous homomorphism $\varphi : \mathbf{R} \to G/H$. Show that there exists a continuous homomorphism $f : \mathbf{R} \to G$ such that $f(\mathbf{R}) \cap H = \{ e \}$ and $G = H \cdot f(\mathbf{R})$. [Reduce to case a) by considering the least upper bound on $G$ of the given topology and the topology for which a fundamental system of neighbourhoods of $o$ is formed by the $\overline{p}^{-1}(\varphi(I))$, where $p : G \to G/H$ is the canonical mapping and $I$ runs through a fundamental system of neighbourhoods of $o$ in $\mathbf{R}$; show that $G$ is locally compact in this new topology (cf. Chapter III, § 4, Exercise 10 e)).]

c) Give an example where the restriction of $p$ to $f(\mathbf{R})$ is not bicontinuous [cf. § 1, Exercise 2f)].