Squares of Menger-bounded groups
Authors:
Michał Machura, Saharon Shelah and Boaz Tsaban
Journal:
Trans. Amer. Math. Soc. 362 (2010), 1751-1764
MSC (2000):
Primary 54H11, 54C65, 03E17
DOI:
https://doi.org/10.1090/S0002-9947-09-05169-1
Published electronically:
November 16, 2009
MathSciNet review:
2574876
Full-text PDF Free Access
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.
- Liljana Babinkostova, Metrizable groups and strict $o$-boundedness, Mat. Vesnik 58 (2006), no. 3-4, 131–138. MR 2318228
- L. Babinkostova, Lj. D. R. Kočinac, and M. Scheepers, Combinatorics of open covers. XI. Menger- and Rothberger-bounded groups, Topology Appl. 154 (2007), no. 7, 1269–1280. MR 2310460, DOI https://doi.org/10.1016/j.topol.2005.09.013
- Reinhold Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), no. 1, 68–122. MR 1545974, DOI https://doi.org/10.1215/S0012-7094-37-00308-9
- Taras Banakh, On index of total boundedness of (strictly) $o$-bounded groups, Topology Appl. 120 (2002), no. 3, 427–439. MR 1897272, DOI https://doi.org/10.1016/S0166-8641%2801%2900084-0
- Taras Banakh, Peter Nickolas, and Manuel Sanchis, Filter games and pathological subgroups of a countable product of lines, J. Aust. Math. Soc. 81 (2006), no. 3, 321–350. MR 2300160, DOI https://doi.org/10.1017/S1446788700014348
- T. Banakh and L. Zdomsky, Selection principles and infinite games on multicovered spaces and their applications, book in progress.
- Taras Banakh and Lyubomyr Zdomskyy, Selection principles and infinite games on multicovered spaces, Selection principles and covering properties in topology, Quad. Mat., vol. 18, Dept. Math., Seconda Univ. Napoli, Caserta, 2006, pp. 1–51. MR 2395750
- 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.
- A. R. Blass, Abelian Group Theory papers, http://www.math.lsa.umich.edu/˜ablass/abgp.html
- Constancio Hernández, Topological groups close to being $\sigma $-compact, Topology Appl. 102 (2000), no. 1, 101–111. MR 1739266, DOI https://doi.org/10.1016/S0166-8641%2898%2900129-1
- 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, DOI https://doi.org/10.4995/agt.2000.3022
- W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Mathematische Zeitschrift 24 (1925), 401–421.
- W. Hurewicz, Über Folgen stetiger Funktionen, Fundamenta Mathematicae 9 (1927), 193–204.
- 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, DOI https://doi.org/10.1016/S0166-8641%2896%2900075-2
- Ljubiša D. R. Kočinac, Selection principles in uniform spaces, Note Mat. 22 (2003/04), no. 2, 127–139. MR 2112735
- Ljubiša D. R. Kočinac, Selected results on selection principles, Proceedings of the 3rd Seminar on Geometry & Topology, Azarb. Univ. Tarbiat Moallem, Tabriz, 2004, pp. 71–104. MR 2090207
- 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
- Adam Krawczyk and Henryk Michalewski, An example of a topological group, Topology Appl. 127 (2003), no. 3, 325–330. MR 1941171, DOI https://doi.org/10.1016/S0166-8641%2802%2900096-2
- Michał Machura and Boaz Tsaban, The combinatorics of the Baer-Specker group, Israel J. Math. 168 (2008), 125–151. MR 2448054, DOI https://doi.org/10.1007/s11856-008-1060-8
- K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte der Wiener Akademie 133 (1924), 421–444.
- H. Michalewski, Function spaces with topology of pointwise convergence, doctoral dissertation, Faculty of Mathematics, Informatics and Mechanic, Warsaw University (2003).
- Heike Mildenberger, Cardinal characteristics for Menger-bounded subgroups, Topology Appl. 156 (2008), no. 1, 130–137. MR 2463833, DOI https://doi.org/10.1016/j.topol.2008.04.011
- H. Mildenberger and S. Shelah, Menger-bounded subgroups of the Baer-Speker group, unpublished notes.
- F. Rothberger, Eine Verschärfung der Eigenschaft C, Fundamenta Mathematicae 30 (1938), 50–55.
- Marion Scheepers, Combinatorics of open covers. I. Ramsey theory, Topology Appl. 69 (1996), no. 1, 31–62. MR 1378387, DOI https://doi.org/10.1016/0166-8641%2895%2900067-4
- Marion Scheepers, Selection principles and covering properties in topology, Note Mat. 22 (2003/04), no. 2, 3–41. MR 2112729
- Ernst Specker, Additive Gruppen von Folgen ganzer Zahlen, Portugal. Math. 9 (1950), 131–140 (German). MR 39719
- Mikhail Tkačenko, Introduction to topological groups, Topology Appl. 86 (1998), no. 3, 179–231. MR 1623960, DOI https://doi.org/10.1016/S0166-8641%2898%2900051-0
- Boaz Tsaban, Some new directions in infinite-combinatorial topology, Set theory, Trends Math., Birkhäuser, Basel, 2006, pp. 225–255. MR 2267150, DOI https://doi.org/10.1007/3-7643-7692-9_7
- Boaz Tsaban, $o$-bounded groups and other topological groups with strong combinatorial properties, Proc. Amer. Math. Soc. 134 (2006), no. 3, 881–891. MR 2180906, DOI https://doi.org/10.1090/S0002-9939-05-08034-2
- Boaz Tsaban and Lyubomyr Zdomskyy, Scales, fields, and a problem of Hurewicz, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 3, 837–866. MR 2421163, DOI https://doi.org/10.4171/JEMS/132
- Tomasz Weiss, A note on unbounded strongly measure zero subgroups of the Baer-Specker group, Topology Appl. 156 (2008), no. 1, 138–141. MR 2463834, DOI https://doi.org/10.1016/j.topol.2008.03.026
- L. S. Zdomskyy, $o$-boundedness of free objects over a Tychonoff space, Mat. Stud. 25 (2006), no. 1, 10–28 (English, with English and Russian summaries). MR 2254996
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 54H11, 54C65, 03E17
Retrieve articles in all journals with MSC (2000): 54H11, 54C65, 03E17
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
MR Author ID:
160185
ORCID:
0000-0003-0462-3152
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
MR Author ID:
632515
Email:
tsaban@math.biu.ac.il
Received by editor(s):
May 1, 2007
Published electronically:
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
Article copyright:
© Copyright 2009
American Mathematical Society
