TY - JOUR

T1 - o-Bounded groups and other topological groups with strong combinatorial properties

AU - Tsaban, Boaz

PY - 2006/3

Y1 - 2006/3

N2 - We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of ℝ (thus strictly o-bounded) which have the Menger and Hurewicz properties but are not σ-compact, and show that the product of two o-bounded subgroups of ℝN may fail to be o-bounded, even when they satisfy the stronger property S1(BΩ, BΩ)- 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 ω-cover of G contains a γ-cover of G.

AB - We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of ℝ (thus strictly o-bounded) which have the Menger and Hurewicz properties but are not σ-compact, and show that the product of two o-bounded subgroups of ℝN may fail to be o-bounded, even when they satisfy the stronger property S1(BΩ, BΩ)- 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 ω-cover of G contains a γ-cover of G.

KW - Luzin sets

KW - Selection principles

KW - o-bounded groups

KW - γ-sets

UR - http://www.scopus.com/inward/record.url?scp=33644763806&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-05-08034-2

DO - 10.1090/S0002-9939-05-08034-2

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AN - SCOPUS:33644763806

SN - 0002-9939

VL - 134

SP - 881

EP - 891

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 3

ER -