Exercise 2, § 1

Let $\mathscr{G}$ be a Hausdorff topology on $\mathbf{R}$, compatible with the group structure of $\mathbf{R}$ and strictly coarser than the usual topology $\mathscr{T}_0$.

a) Show that every open neighbourhood of $o$ with respect to $\mathscr{T}$ is unbounded in $\mathbf{R}$ (note that a Hausdorff topology which is coarser than the topology of a compact space must coincide with the latter).

b) Let $V \neq \mathbf{R}$ be a symmetric open neighbourhood of $o$ with respect to $\mathscr{G}$, and let $W$ be a symmetric open neighbourhood of $o$ with respect to $\mathscr{G}$ such that $W + W \subset V$. Show that if $a$ is the length of the component of $V$ (with respect to $\mathscr{G}_0$) which contains $o$, then every component (with respect to $\mathscr{G}_0$) of $W$ has length $\leq a$. Moreover, the set of lengths of components of the interior of $\mathbf{R} - W$ (with respect to $\mathscr{G}_0$) is bounded [use a) and the fact that there exists a symmetric open neighbourhood $W_1$ of $o$ with respect to $\mathscr{T}$ such that $W_1 + W_1 \subset W$].

c) Deduce from b) that $\mathbf{R}$ is precompact and not locally compact in the topology $\mathscr{G}$.

d) Show likewise that $\mathbf{Z}$ is precompact and not locally compact in any topology $\mathscr{G}$ compatible with its group structure and distinct from the discrete topology.

e) Let $f$ be a continuous homomorphism of a subgroup $\Gamma$ of $\mathbf{R}$, not consisting of $o$ alone and endowed with the topology induced by $\mathscr{G}_0$, into a complete Hausdorff group $G$. Show that if $f$ is not an isomorphism of $\Gamma$ onto the subgroup $f(\Gamma')$ of $G$, then $f(\Gamma')$ is relatively compact in $G$ [use c) and d)].

\* $f$) For every integer $n \geq 2$, give examples of continuous injective homomorphisms $f$ of $\mathbf{R}$ into $\mathbf{T}^n$ such that $f(\mathbf{R})$ is dense in $\mathbf{T}^n$ (cf. Chapter VII, § 1, no. 4, Corollary 1 of Proposition 7). \*