Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters

TL;DR

This paper classifies certain countably compact Abelian groups of size continuum using selective ultrafilters, revealing diverse topological structures and their algebraic properties under set-theoretic assumptions.

Contribution

It provides a classification of non-torsion Abelian groups of size continuum with countably compact topologies based on selective ultrafilters, introducing new topological constructions.

Findings

Existence of multiple non-homeomorphic countably compact topologies for these groups.

Every Abelian group of size at most continuum is algebraically countably compact.

Consistency results on the existence of topologies without non-trivial convergent sequences.

Abstract

Assuming the existence of $\mathfrak c$ incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality $\mathfrak c$ that admit a countably compact group topology. We show that for each $\kappa \in [\mathfrak c, 2^\mathfrak c]$ each of these groups has a countably compact group topology of weight $\kappa$ without non-trivial convergent sequences and another that has convergent sequences. Assuming the existence of $2^\mathfrak c$ selective ultrafilters, there are at least $2^\mathfrak c$ non homeomorphic such topologies in each case and we also show that every Abelian group of cardinality at most $2^\mathfrak c$ is algebraically countably compact. We also show that it is consistent that every Abelian group of cardinality $\mathfrak c$ that admits a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality.