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

Abstract

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