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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. |
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