## Abstract

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact) if there is a sequence u=(u_{n}) in G such that τ is the finest precompact group topology on G making u=(u_{n}) converge to zero. It is proved that a metrizable precompact abelian group (G, τ) is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ and the groups (G, τ) and (G, η) have the same Pontryagin dual groups (in other words, (G, τ) is not a Mackey group in the class of maximally almost periodic groups).We give a complete description of all ss-precompact abelian groups modulo countable ss-precompact groups from which we derive:. (1)No infinite pseudocompact abelian group is ss-precompact.(2)An ss-precompact group G is a k-space if and only if G is countable and sequential.(3)An ss-precompact group is hereditarily disconnected.(4)An ss-precompact group has countable tightness. We provide also a description of the sequentially complete ss-precompact abelian groups.

Original language | English |
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Pages (from-to) | 505-519 |

Number of pages | 15 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 412 |

Issue number | 1 |

DOIs | |

State | Published - 1 Apr 2014 |

## Keywords

- B-embedded subgroup
- Characterized subgroup
- Characterizing sequence
- Finest precompact extension
- Precompact group topology
- T-sequence
- TB-sequence