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

2020/06/17 by Matheus Koveroff Bellini, Vinicius de Oliveira Rodrigues, Bellini, Matheus Koveroff +3
Computer Science · Mathematics · #22A05 #54G20 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Primary 54H11 #Rings, Modules, and Algebras #Secondary 54A35 #math.GN #msc:22A05 #msc:54A35 #msc:54G20 #msc:54H11

paper · pdf · doi:10.48550/arxiv.2006.09961

arXiv admin note: text overlap with arXiv:1909.03340

arxiv created 2020/06/17 · openalex publication_date 2020/06/17 · arxiv updated 2020/06/24 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We force a classification of all the Abelian groups of cardinality at most 2^\mathfrak c that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and Shakhmatov for cardinality at most 2^{\mathfrak c}, by showing that if a non-torsion Abelian group of size at most 2^\mathfrak c admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.

Citations

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

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