-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

DOI:
https://doi.org/10.1090/S0002-9939-05-08034-2

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 (thus strictly -bounded) which have the Menger and Hurewicz properties but are not -compact, and show that the product of two -bounded subgroups of may fail to be -bounded, even when they satisfy the stronger property . 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 of size continuum such that every countable Borel -cover of contains a -cover of .

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Additional Information

**Boaz Tsaban**

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

Email:
boaz.tsaban@weizmann.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-05-08034-2

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.