Exercise 3, § 1

Show that the group $\mathbf{T}$ is algebraically isomorphic to the product $\mathbf{R} \times (\mathbf{Q}/\mathbf{Z})$ (take a suitable Hamel base in $\mathbf{R}$). Deduce that there exists a Hausdorff topology on $\mathbf{R}$, compatible with its group structure, not comparable with the usual topology, and with respect to which $\mathbf{R}$ is precompact.