Algebraic structure of countably compact nontorsion Abelian groups of size continuum from selective ultrafilters
Abstract
Assuming the existence of $\mathfrak c$ incomparable selective ultrafilters, we classify the nontorsion 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 nontrivial 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 nontrivial convergent sequences whose weight has countable cofinality.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.03340
 Bibcode:
 2019arXiv190903340B
 Keywords:

 Mathematics  General Topology
 EPrint:
 new revised and expanded version that includes new coauthors