Squares of Menger-bounded groups

Machura, Michał; Shelah, Saharon; Tsaban, Boaz

doi:10.1090/S0002-9947-09-05169-1

Squares of Menger-bounded groups
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by Michał Machura, Saharon Shelah and Boaz Tsaban PDF

Trans. Amer. Math. Soc. 362 (2010), 1751-1764 Request permission

Abstract:

Using a portion of the Continuum Hypothesis, we prove that there is a Menger-bounded (also called $o$-bounded) subgroup of the Baer-Specker group $\mathbb {Z}^{\mathbb {N}}$, whose square is not Menger-bounded. This settles a major open problem concerning boundedness notions for groups and implies that Menger-bounded groups need not be Scheepers-bounded. This also answers some questions of Banakh, Nickolas, and Sanchis.

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