Menger-bounded groups and axioms about filters

Jialiang He; Boaz Tsaban; Shuguo Zhang

doi:10.1016/j.topol.2021.107914

Menger-bounded groups and axioms about filters

Jialiang He, Boaz Tsaban, Shuguo Zhang

Department of Mathematics

Sichuan University

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

A topological group G is Menger-bounded if, for each sequence U₁,U₂,… of open sets, there are finite sets F₁,F₂,… such that G=⋃_nF_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
Article number	107914
Journal	Topology and its Applications
Volume	309
DOIs	https://doi.org/10.1016/j.topol.2021.107914
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
National Natural Science Foundation of China	11771311, 11801386
Council for Higher Education

Keywords

Menger-bounded set
NCF
Rapid filter
Scheepers-bounded set

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10.1016/j.topol.2021.107914

Cite this

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AB - A topological group G is Menger-bounded if, for each sequence U1,U2,… of open sets, there are finite sets F1,F2,… such that G=⋃nFn⋅Un. 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 ZN 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.

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Menger-bounded groups and axioms about filters

Abstract

Bibliographical note

Funding

Keywords

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