Countably compact group topologies on torsion-free Abelian groups

Abstract

This work presents advancements obtained in consistency results on the field of topological algebra, especially concerning countably compact group topologies and whether they may contain non-trivial convergent sequences. Furthering the methods and techniques already established in this line of research, we have obtained the following results, the first two of which already published in international journals with peer arbitration: first, obtain p-compact group topologies on arbitrarily large torsion-free Abelian groups without non-trivial convergent sequences, for p a selective ultrafilter; second, obtain group topologies on arbitrarily large free Abelian groups without non-trivial convergent sequences all of whose finite powers are countably compact, assuming c incomparable selective ultrafilters; third, a forcing model in which a torsion-free Abelian group whose cardinality is countably cofinal admits a p-compact group topology for p a selective ultrafilter. These results improve upon previously established theory and showcase the first consistent examples regarding the properties of -compactness and arbitrarily largeness in their respective settings.