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

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.

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