Exercise 3, § 1

For every real number $\alpha > 0$, let $G_\alpha$ be the topological group $\mathbf{R}/\alpha \mathbf{Z}$, and let $G$ be the topological product group $\prod_\alpha G_\alpha$ ($\alpha$ varying in the set of number $> 0$). For every $x \in \mathbf{R}$, let $t_\alpha(x)$ be the canonical image of $x$ in $G_\alpha$; the mapping $\phi : x \to (t_\alpha(x))$ is a continuous injective homomorphism of $\mathbf{R}$ in $G$. We consider on $\mathbf{R}$ the topology that is the inverse image by $\phi$ of that of $G$, and denote by $E$ the topological group formed by $\mathbf{R}$ with this topology. Show that when $E$ is considered as a vector space over $\mathbf{R}$ its topology satisfies (EVT'_I) and (EVT'_II) but not (EVT'_III).