-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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 .

**1.**L. Babinkostova, Lj. D. R. Kocinac, and M. Scheepers,*Combinatorics of open covers (XI): Topological groups*, in progress.L. Babinkostova, Lj. D. R. Kocinac, and M. Scheepers,*Combinatorics of open covers (XI): Topological groups*, in progress.**2.**T. Banakh, P. Nickolas, and M. Sanchis,*Filter games and pathological subgroups of a countable product of lines*, Journal of the Australian Mathematical Society, to appear.**3.**Tomek Bartoszyński,*Combinatorial aspects of measure and category*, Fund. Math.**127**(1987), no. 3, 225–239. MR**917147****4.**Tomek Bartoszynski, Saharon Shelah, and Boaz Tsaban,*Additivity properties of topological diagonalizations*, J. Symbolic Logic**68**(2003), no. 4, 1254–1260. MR**2017353**, 10.2178/jsl/1067620185**5.**T. Bartoszynski and B. Tsaban,*Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures*, Proceedings of the American Mathematical Society, to appear.**6.**Lev Bukovský, Ireneusz Recław, and Miroslav Repický,*Spaces not distinguishing convergences of real-valued functions*, Topology Appl.**112**(2001), no. 1, 13–40. MR**1815270**, 10.1016/S0166-8641(99)00226-6**7.**Fred Galvin and Arnold W. Miller,*𝛾-sets and other singular sets of real numbers*, Topology Appl.**17**(1984), no. 2, 145–155. MR**738943**, 10.1016/0166-8641(84)90038-5**8.**J. Gerlits and Zs. Nagy,*Some properties of 𝐶(𝑋). I*, Topology Appl.**14**(1982), no. 2, 151–161. MR**667661**, 10.1016/0166-8641(82)90065-7**9.**Constancio Hernández,*Topological groups close to being 𝜎-compact*, Topology Appl.**102**(2000), no. 1, 101–111. MR**1739266**, 10.1016/S0166-8641(98)00129-1**10.**C. Hernández, D. Robbie, and M. Tkachenko,*Some properties of 𝑜-bounded and strictly 𝑜-bounded groups*, Appl. Gen. Topol.**1**(2000), no. 1, 29–43. MR**1796930****11.**Winfried Just, Arnold W. Miller, Marion Scheepers, and Paul J. Szeptycki,*The combinatorics of open covers. II*, Topology Appl.**73**(1996), no. 3, 241–266. MR**1419798**, 10.1016/S0166-8641(96)00075-2**12.**Adam Krawczyk and Henryk Michalewski,*An example of a topological group*, Topology Appl.**127**(2003), no. 3, 325–330. MR**1941171**, 10.1016/S0166-8641(02)00096-2**13.**A. Krawczyk and H. Michalewski,*Linear metric spaces close to being -compact*, Technical Report 46 (2001) of the Institute of Mathematics, Warsaw University.`http://www.minuw.edu.pl/english/research/reports/tr-imat/46/products.ps`**14.**Arnold W. Miller,*A nonhereditary Borel-cover 𝛾-set*, Real Anal. Exchange**29**(2003/04), no. 2, 601–606. MR**2083799****15.**Marion Scheepers,*Combinatorics of open covers. I. Ramsey theory*, Topology Appl.**69**(1996), no. 1, 31–62. MR**1378387**, 10.1016/0166-8641(95)00067-4**16.**M. Scheepers,*Selection principles and covering properties in topology*, Note di Matematica**22**(2003), 3-41.**17.**Marion Scheepers and Boaz Tsaban,*The combinatorics of Borel covers*, Topology Appl.**121**(2002), no. 3, 357–382. MR**1908999**, 10.1016/S0166-8641(01)00078-5**18.**Mikhail Tkačenko,*Introduction to topological groups*, Topology Appl.**86**(1998), no. 3, 179–231. MR**1623960**, 10.1016/S0166-8641(98)00051-0**19.**B. Tsaban,*Selection principles in Mathematics: A milestone of open problems*, Note di Matematica**22**(2003), 179-208.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
54H11,
37F20

Retrieve articles in all journals with MSC (2000): 54H11, 37F20

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:
http://dx.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.