Countably compact group topologies on non-torsion abelian groups of size continuum with non-trivial convergent sequences

TL;DR

This paper proves that non-torsion abelian groups of size continuum with a countably compact Hausdorff topology can be endowed with such a topology that also contains non-trivial convergent sequences, under certain set-theoretic assumptions.

Contribution

It establishes the existence of countably compact group topologies with non-trivial convergent sequences on non-torsion abelian groups of continuum size, answering a specific open question.

Findings

Countably compact topologies with non-trivial convergent sequences exist under $rak{p} = rak{c}$.

The result applies to non-torsion abelian groups of size continuum.

It extends previous understanding of compactness properties in topological groups.

Abstract

Under $\mathfrak{p} = \mathfrak{c}$, we answer Question 24 of \cite{dikranjan&shakhmatov3} for cardinality ${\mathfrak c}$ , by showing that if a non-torsion Abelian group of size continuum admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.