Topological classification of zero-dimensional $M_omega$-groups

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 $Ucap K$ is open in $K$ for every $KinK$. It is shown that any two non-metrizable uncountable separable zero-dimenisional $M_omega$-groups are homeomorphic. Together with Zelenyuks 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 $G$.