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$o$-bounded groups and other topological groups with strong combinatorial properties

Author: Boaz Tsaban
Journal: Proc. Amer. Math. Soc. 134 (2006), 881-891
MSC (2000): Primary 54H11; Secondary 37F20
Published electronically: July 7, 2005
MathSciNet review: 2180906
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Abstract: We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of $\mathbb{R} $(thus strictly $o$-bounded) which have the Menger and Hurewicz properties but are not $\sigma$-compact, and show that the product of two $o$-bounded subgroups of $\mathbb{R} ^{\mathbb{N} }$ may fail to be $o$-bounded, even when they satisfy the stronger property $\mathsf{S}_1(\mathcal{B}_{\Omega},\mathcal{B}_{\Omega})$. This solves a problem of Tkacenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups $G$ of size continuum such that every countable Borel $\omega$-cover of $G$ contains a $\gamma$-cover of $G$.

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Additional Information

Boaz Tsaban
Affiliation: Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel

Keywords: $o$-bounded groups, $\gamma$-sets, Luzin sets, selection principles.
Received by editor(s): July 8, 2003
Received by editor(s) in revised form: September 20, 2004
Published electronically: July 7, 2005
Communicated by: Alan Dow
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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