## Abstract

A sequence [u_{n}]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 lim_{n} u_{n} = 0. We say that a subgroup H of an infinite compact Abelian group X is T-characterized if there is a T-sequence u = [u_{n}] in the dual group of X, such that H = [x ∈ X: (u_{n};, 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 F_{σ}-subgroup of X, that gives a negative answer to Problem 3.3 in Dikranjan and Gabriyelyan (Topol. Appl. 2013, 160, 2427-2442).

Original language | English |
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Pages (from-to) | 194-212 |

Number of pages | 19 |

Journal | Axioms |

Volume | 4 |

Issue number | 2 |

DOIs | |

State | Published - 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