The space of subgroups of an abelian group

TL;DR

This paper analyzes the topological structure of the space of all subgroups of countable abelian groups using Cantor-Bendixson analysis, leading to a complete classification of these spaces up to homeomorphism.

Contribution

It provides a comprehensive classification of the topological spaces formed by subgroups of countable abelian groups, a novel application of Cantor-Bendixson analysis.

Findings

Complete classification of subgroup spaces up to homeomorphism

Identification of topological types for different abelian groups

Application of Cantor-Bendixson analysis to algebraic structures

Abstract

We carry out the Cantor-Bendixson analysis of the space of all subgroups of any countable abelian group and we deduce a complete classification of such spaces up to homeomorphism.