On T-characterized subgroups of compact Abelian groups

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4 Scopus citations

Abstract

A sequence [un]n∈ωin abstract additively-written Abelian group G is called a T-sequence if there is a Hausdorff group topology on G relative to which limn un = 0. We say that a subgroup H of an infinite compact Abelian group X is T-characterized if there is a T-sequence u = [un] in the dual group of X, such that H = [x ∈ X: (un;, x) → 1]. We show that a closed subgroup H of X is T-characterized if and only if H is a GΔ-subgroup of X and the annihilator of H admits a Hausdorff minimally almost periodic group topology. All closed subgroups of an infinite compact Abelian group X are T-characterized if and only if X is metrizable and connected. We prove that every compact Abelian group X of infinite exponent has a T-characterized subgroup, which is not an Fσ-subgroup of X, that gives a negative answer to Problem 3.3 in Dikranjan and Gabriyelyan (Topol. Appl. 2013, 160, 2427-2442).

Original languageEnglish
Pages (from-to)194-212
Number of pages19
JournalAxioms
Volume4
Issue number2
DOIs
StatePublished - 1 Jun 2015

Keywords

  • Characterized subgroup
  • Dual group
  • T-characterized subgroup
  • T-sequence
  • Von Neumann radical

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Mathematical Physics
  • Logic
  • Geometry and Topology

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