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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Squares of Menger-bounded groups

Author(s): Michał Machura; Saharon Shelah; Boaz Tsaban
Journal: Trans. Amer. Math. Soc. 362 (2010), 1751-1764.
MSC (2000): Primary 54H11, 54C65, 03E17
Posted: November 16, 2009
MathSciNet review: 2574876
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Abstract | References | Similar articles | Additional information

Abstract: Using a portion of the Continuum Hypothesis, we prove that there is a Menger-bounded (also called $ o$-bounded) subgroup of the Baer-Specker group $ \mathbb{Z}^{\mathbb{N}}$, whose square is not Menger-bounded. This settles a major open problem concerning boundedness notions for groups and implies that Menger-bounded groups need not be Scheepers-bounded. This also answers some questions of Banakh, Nickolas, and Sanchis.

References:

1.
L. Babinkostova, Metrizable groups and strict $ o$-boundedness, Matematicki Vesnik 58 (2006), 131-138. MR 2318228 (2008b:54055)

2.
L. Babinkostova, Lj. D.R. Kočinac, and M. Scheepers, Combinatorics of open covers (XI): Menger- and Rothberger-bounded groups, Topology and its Applications 154 (2007), 1269-1280. MR 2310460 (2008g:54051)

3.
R. Baer, Abelian groups without elements of finite order, Duke Mathematical Journal 3 (1937), 68-122. MR 1545974

4.
T. Banakh, On index of total boundedness of (strictly) $ o$-bounded groups, Topology and its Applications 120 (2002), 427-439. MR 1897272 (2003c:22003)

5.
T. Banakh, P. Nickolas, and M. Sanchis, Filter games and pathological subgroups of a countable product of lines, Journal of the Australian Mathematical Society 81 (2006), 321-350. MR 2300160 (2008c:54024)

6.
T. Banakh and L. Zdomsky, Selection principles and infinite games on multicovered spaces and their applications, book in progress.

7.
T. Banakh and L. Zdomskyy, Selection principles and infinite games on multicovered spaces in: Selection Principles and Covering Properties in Topology (Lj. D.R. Kočinac, ed.), Quaderni di Matematica 18 (2006), Seconda Universita di Napoli, Caserta, 2-51. MR 2395750 (2009g:54047)

8.
A. R. Blass, Combinatorial cardinal characteristics of the continuum, in: Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), Kluwer Academic Publishers, Dordrecht, to appear.

9.
A. R. Blass, Abelian Group Theory papers, http://www.math.lsa.umich.edu/ ˜ablass/abgp.html

10.
C. Hernandez, Topological groups close to being $ \sigma$-compact, Topology and its Applications 102 (2000), 101-111. MR 1739266 (2000k:54032)

11.
C. Hernandez, D. Robbie and M. Tkačenko, Some properties of $ o$-bounded and strictly $ o$-bounded groups, Applied General Topology 1 (2000), 29-43. MR 1796930 (2001g:22002)

12.
W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Mathematische Zeitschrift 24 (1925), 401-421.

13.
W. Hurewicz, Über Folgen stetiger Funktionen, Fundamenta Mathematicae 9 (1927), 193-204.

14.
W. Just, A. Miller, M. Scheepers, and P. Szeptycki, The combinatorics of open covers. II, Topology and its Applications 73 (1996), 241-266. MR 1419798 (98g:03115a)

15.
L. Kočinac, Selection principles in uniform spaces, Note di Matematica 22 (2003), 127-139. MR 2112735 (2006b:54019)

16.
L. Kocinac, Selected results on selection principles, in: Proceedings of the 3rd Seminar on Geometry and Topology (Sh. Rezapour, ed.), July 15-17, Tabriz, Iran, 2004, 71-104. MR 2090207 (2005g:54038)

17.
A. Krawczyk and H. Michalewski, Linear metric spaces close to being $ \sigma$-compact, Technical Report 46 (2001) of the Institute of Mathematics, Warsaw University. www.minuw.edu.pl/ english/research/reports/ tr-imat/46/products.ps

18.
A. Krawczyk and H. Michalewski, An example of a topological group, Topology and its Applications 127 (2003), 325-330. MR 1941171 (2003j:54034)

19.
M. Machura and B. Tsaban, The combinatorics of the Baer-Specker group, Israel Journal of Mathematics, 168 (2008), 125-151. MR 2448054 (2009g:20116)

20.
K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte der Wiener Akademie 133 (1924), 421-444.

21.
H. Michalewski, Function spaces with topology of pointwise convergence, doctoral dissertation, Faculty of Mathematics, Informatics and Mechanic, Warsaw University (2003).

22.
H. Mildenberger, Cardinal characteristics for Menger-bounded subgroups, Topology and its Applications 156 (2008), 130-137. MR 2463833 (2009i:03049)

23.
H. Mildenberger and S. Shelah, Menger-bounded subgroups of the Baer-Speker group,

unpublished notes.

24.
F. Rothberger, Eine Verschärfung der Eigenschaft C, Fundamenta Mathematicae 30 (1938), 50-55.

25.
M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology and its Applications 69 (1996), 31-62. MR 1378387 (97h:90123)

26.
M. Scheepers, Selection principles and covering properties in topology, Note di Matematica 22 (2003), 3-41. MR 2112729 (2005h:54024)

27.
E. Specker, Additive Gruppen von Folgen ganzer Zahlen, Portugaliae Mathematica 9 (1950), 131-140. MR 0039719 (12:587b)

28.
M. Tkačenko, Introduction to topological groups, Topology and its Applications 86 (1998), 179-231. MR 1623960 (99b:54064)

29.
B. Tsaban, Some new directions in infinite-combinatorial topology, in: Set Theory (J. Bagaria and S. Todorčevic, eds.), Trends in Mathematics, Birkhauser, 2006, 225-255. MR 2267150 (2007f:03064)

30.
B. Tsaban, $ o$-bounded groups and other topological groups with strong combinatorial properties, Proceedings of the American Mathematical Society 134 (2006), 881-891. MR 2180906 (2006h:54034)

31.
B. Tsaban and L. Zdomskyy, Scales, fields, and a problem of Hurewicz, Journal of the European Mathematical Society (JEMS) 10 (2008), 837-866. MR 2421163

32.
T. Weiss, A note on unbounded strongly measure zero subgroups of the Baer-Specker group, Topology and its Applications 156 (2008), 138-141. MR 2463834

33.
L. Zdomskyy, $ o$-Boundedness of free objects over a Tychonoff space, Matematychni Studii 25 (2006), 10-28. MR 2254996 (2008a:54025)

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Additional Information:

Michał Machura
Affiliation: Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland - and - Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel
Email: machura@ux2.math.us.edu.pl

Saharon Shelah
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel - and - Department of Mathematics, Rutgers University, New Brunswick, Piscataway, New Jersey 08854
Email: shelah@math.huji.ac.il

Boaz Tsaban
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel - and - Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Email: tsaban@math.biu.ac.il

DOI: 10.1090/S0002-9947-09-05169-1
PII: S 0002-9947(09)05169-1
Received by editor(s): May 1, 2007
Posted: November 16, 2009
Additional Notes: The authors were partially supported by the EU Research and Training Network HPRN-CT-2002-00287, United States-Israel BSF Grant 2002323, and the Koshland Center for Basic Research, respectively
Copyright of article: Copyright 2009, American Mathematical Society




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