The Protasov-Zelenyuk topology and ideal convergence

2025/12/03 by Außenhofer, Lydia, Dikranjan, Dikran, Bruno, Anna Giordano
#11B05 #20K45 #22A05 #22C05 #54H11 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2512.03885

Abstract

The so-called T-sequences \mathbf u in a group G, and the related finest Hausdorff group topology T_\mathbf u on G that makes \mathbf u a null sequence, were introduced by Protasov and Zelenyuk 35 years ago and since then they became a fundamental tool in the field of topological groups. More recently, in the abelian case, the subfamily of T-sequences called TB-sequences was introduced, as well as the finest precompact group topology T_pu that makes \mathbf u a null sequence. Here we study the counterpart of all these notions with respect to ideal convergence in place of the classical notion of convergence of a sequence. Also, we study their relation to the already established field of I-characterized subgroups of compact abelian groups.

Citations

• Characterizing sequences for precompact group topologies
• On T-sequences and characterized subgroups

Related

• Coarse structures on groups defined by T-sequences
• Topologies on groups determined by sets of convergent sequences
• Group topologies on integers and S-unit equations
• On convergent sequences in dual groups
• On T-sequences and characterized subgroups