  A topological group $G$ is called an $M_\omega$-group if it admits a
countable cover $\K$ by closed metrizable subspaces of $G$ such that a subset
$U$ of $G$ is open in $G$ if and only if $U\cap K$ is open in $K$ for every
$K\in\K$. It is shown that any two non-metrizable uncountable separable
zero-dimenisional $M_\omega$-groups are homeomorphic. Together with Zelenyuk's
classification of countable $k_\omega$-groups this implies that the topology of
a non-metrizable zero-dimensional $M_\omega$-group $G$ is completely determined
by its density and the compact scatteredness rank $r(G)$ which, by definition,
is equal to the least upper bound of scatteredness indices of scattered compact
subspaces of $G$.
