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

2019/09/07 by M. K. Bellini, A. C. Boero, Bellini, M. K. +5 · 1 citation
Mathematics · #FOS: Mathematics #General Topology (math.GN) #math.GN

paper · pdf · doi:10.48550/arxiv.1909.03340

new revised and expanded version that includes new co-authors

arxiv created 2021/04/22 · arxiv updated 2021/04/26

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 κ∈ [\mathfrak c, 2^\mathfrak c] 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 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.

Cited by

• Forcing a classification of non-torsion Abelian groups of size at most 2^\mathfrak c with non-trivial convergent sequences

Related

• U-compact group topologies without convergent sequences on countably cofinal torsion-free Abelian groups
• Forcing a classification of non-torsion Abelian groups of size at most 2^\mathfrak c with non-trivial convergent sequences
• Countably compact group topologies on arbitrarily large free Abelian groups
• Countably compact group topologies on non-torsion abelian groups of size continuum with non-trivial convergent sequences
• On p-compact group topologies on direct sums of \mathbb Q