$\mathcal{U}$-compact group topologies without convergent sequences on countably cofinal torsion-free Abelian groups

TL;DR

This paper constructs a consistent example of a torsion-free Abelian group with a Hausdorff topology that is $-compact, contains no non-trivial convergent sequences, and is built using forcing, addressing a previously open question.

Contribution

It provides a forcing-based construction of a $$-compact topology on a countably cofinal torsion-free Abelian group without convergent sequences, answering an open problem.

Findings

Existence of a $$-compact topology on $Q^{()}$ without convergent sequences

Use of forcing to demonstrate consistency of such topologies

Addresses a question from prior research (arXiv:1904.05928)

Abstract

We obtain a forcing construction that shows that it is consistent that the torsion-free Abelian group $\mathbb{Q}^{(\lambda)}$ admits a Hausdorff group topology which is also $\mathcal{U}$-compact and contains no non-trivial convergent sequences, where $\lambda$ is a cardinal whose cofinality is $\omega$ and $\mathcal{U}$ is a selective ultrafilter. This answers a question posed in arXiv:1904.05928.