## Abstract

A topological group G is Menger-bounded if, for each sequence U_{1},U_{2},… of open sets, there are finite sets F_{1},F_{2},… such that G=⋃_{n}F_{n}⋅U_{n}. It is Scheepers-bounded if all of its finite powers are Menger-bounded. A notorious open problem asks whether, consistently, every product of two Menger-bounded subgroups of the Baer–Specker group Z^{N} is Menger-bounded. We prove that the same assertion for Scheepers-bounded groups is equivalent to the set-theoretic axiom NCF (Near Coherence of Filters). We also show that Menger-bounded sets are not productive, and that the preservation of Scheepers-bounded subsets of [N]^{ω} by finite-to-one quotients is equivalent to nonexistence of rapid filters.

Original language | English |
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Article number | 107914 |

Journal | Topology and its Applications |

Volume | 309 |

DOIs | |

State | Published - 15 Mar 2022 |

### Bibliographical note

Publisher Copyright:© 2021 Elsevier B.V.

### Funding

This research is supported by the Israel Council for Higher Education , PBC Fellowship Program for Outstanding Chinese and Indian Post-Doctoral Fellows. The first author is partially supported by a National Science Foundation of China grant # 11801386 . The third author is partially supported by a National Science Foundation of China grant # 11771311 . We thank the referees for their work on the evaluation of this paper.

Funders | Funder number |
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National Natural Science Foundation of China | 11771311, 11801386 |

Council for Higher Education |

## Keywords

- Menger-bounded set
- NCF
- Rapid filter
- Scheepers-bounded set