Menger-bounded groups and axioms about filters

Jialiang He, Boaz Tsaban, Shuguo Zhang

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish
Article number107914
JournalTopology and its Applications
StatePublished - 15 Mar 2022

Bibliographical note

Publisher Copyright:
© 2021 Elsevier B.V.


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.

FundersFunder number
National Natural Science Foundation of China11771311, 11801386
Council for Higher Education


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


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