Topological classification of zero-dimensional $M_\omega$-groups

TL;DR

This paper classifies non-metrizable uncountable separable zero-dimensional $M_$-groups, showing they are all homeomorphic, and determines their topology based on density and scatteredness rank.

Contribution

It provides a topological classification of non-metrizable zero-dimensional $M_$-groups, extending previous work and linking topology to scatteredness rank.

Findings

Any two non-metrizable uncountable separable zero-dimensional $M_$-groups are homeomorphic.

The topology of such groups is determined by their density and scatteredness rank.

Classification aligns with Zelenyuk's results on countable $k_$-groups.

Abstract

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 $G$.