## Abstract

Let X be a compact abelian group. A subgroup H of X is called characterized if there exists a sequence u=(u_{n}) of characters of X such that H=s_{u}(X), where su(X):={x∈X:(un,x)→0 in T}. Every characterized subgroup is an F_{σδ}-subgroup of X. We show that every G_{δ}-subgroup of X is characterized. On the other hand, X has non-characterized F_{σ}-subgroups.A subgroup H of X is said to be countable modulo compact (CMC) if H has a subgroup K such that it is a compact G_{δ}-subgroup of X and H/K is countable. It is proved that every characterized subgroup H of X is CMC if and only if X has finite exponent. This result gives a complete description of the characterized subgroups of compact abelian groups of finite exponent.For every sequence u=(u_{n}) of characters of X we define a refinement X_{u} of X, that is a Čech complete locally quasi-convex (almost metrizable) group. With the sequence u we associate the closed subgroup H_{u} of X_{u} and the natural projection π_{X}:X_{u}→X such that π_{X}(H_{u})=s_{u}(X). This provides a description of the characterized subgroups of arbitrary compact abelian groups, extending the previously existing result from [25]. This description is new even in the case of metrizable compact groups.

Original language | English |
---|---|

Pages (from-to) | 2427-2442 |

Number of pages | 16 |

Journal | Topology and its Applications |

Volume | 160 |

Issue number | 18 |

DOIs | |

State | Published - 1 Dec 2013 |

## Keywords

- Borel hierarchy
- Characterized subgroup
- Characterizing sequence
- T-sequence
- TB-sequence

## ASJC Scopus subject areas

- Geometry and Topology