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
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
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 -