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 $U\cap K$ 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 $G$.

💡 Research Summary

The paper investigates a special class of topological groups called M ω‑groups. By definition, a group G is an M ω‑group if there exists a countable family 𝒦 of closed metrizable subspaces covering G such that a set U⊆G is open precisely when U∩K is open in each K∈𝒦. This condition generalizes the well‑known k ω‑group notion, allowing the global topology to be determined by a countable collection of metrizable pieces.

The authors focus on zero‑dimensional M ω‑groups, i.e., groups whose underlying space is totally disconnected and has a base consisting of clopen sets. Within this setting they prove a striking rigidity theorem: any two non‑metrizable, uncountable, separable, zero‑dimensional M ω‑groups are homeomorphic. The proof proceeds in two main steps. First, they show that such groups are σ‑compact and that every compact subspace is scattered. Using the scatteredness index of a compact scattered space, they introduce the invariant r(G), the compact scatteredness rank of G, defined as the supremum of the scatteredness indices of all scattered compact subsets of G. This invariant captures the “complexity” of the compact part of the group.

Second, they demonstrate that the pair of cardinal invariants (d(G), r(G))—the density of G and its compact scatteredness rank—completely determines the homeomorphism type. They construct a canonical model space, essentially the Cantor cube 2^ℵ₀ equipped with a zero‑dimensional topology, and show that for any zero‑dimensional M ω‑group G there exists a homeomorphism onto a subspace of this model uniquely prescribed by d(G) and r(G). Consequently, two such groups are homeomorphic if and only if they share the same density and the same compact scatteredness rank.

By combining this result with Zelenyuk’s earlier classification of countable k ω‑groups, the paper obtains a full classification of all non‑metrizable zero‑dimensional M ω‑groups: the topology of any such group is completely determined by its density and r(G). The authors also discuss the extremal cases. When r(G)=ω₁ (the first uncountable ordinal), the group attains maximal scattered complexity, while smaller values of r(G) correspond to “intermediate” groups whose structure is closer to that of the classical Cantor set.

The paper concludes with several remarks. The rigidity phenomenon shows that the presence of a countable metrizable cover imposes strong constraints on the global topology, especially under the zero‑dimensional hypothesis, effectively reducing the classification problem to a pair of set‑theoretic invariants. This insight opens avenues for future research: extending the classification to non‑zero‑dimensional M ω‑groups, investigating other types of countable covers, and exploring connections with descriptive set theory and the theory of scattered spaces. Overall, the work provides a clean and elegant answer to the homeomorphism classification of a broad class of non‑metrizable topological groups.