  We prove that a large class of metrizable group topologies for subgroups of
$\mathbb{R}^n$ and the completions of the subgroups are locally isometric to,
respectively, metrizable group topologies for $\mathbb{Z}$ and their
completions, first studied by Nienhuys. This will prove, in particular, that
all the complete groups in question are one dimensional, locally totally
disconnected, and not locally compact. The metrizable topologies on the
subgroups of $\mathbb{R}^n$ are formed by specifying a sequence in
$\mathbb{R}^n$ and the rate at which it must converge to the identity.

General Topology