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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$o$-bounded groups and other topological groups with strong combinatorial properties
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by Boaz Tsaban PDF
Proc. Amer. Math. Soc. 134 (2006), 881-891 Request permission


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 Tkačenko 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
  • MR Author ID: 632515
  • Email:
  • 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
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 881-891
  • MSC (2000): Primary 54H11; Secondary 37F20
  • DOI:
  • MathSciNet review: 2180906