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Linear $ \sigma$-additivity and some applications


Authors: Tal Orenshtein and Boaz Tsaban
Journal: Trans. Amer. Math. Soc. 363 (2011), 3621-3637
MSC (2010): Primary 54D20, 54D80, 54H11; Secondary 03E17, 03E35
DOI: https://doi.org/10.1090/S0002-9947-2011-05228-1
Published electronically: February 7, 2011
MathSciNet review: 2775821
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Abstract: We show that countable increasing unions preserve a large family of well-studied covering properties, which are not necessarily $ \sigma$-additive. Using this, together with infinite-combinatorial methods and simple forcing theoretic methods, we explain several phenomena, settle problems of Just, Miller, Scheepers and Szeptycki (1996), Gruenhage and Szeptycki (2005), Tsaban and Zdomskyy (2008), and Tsaban (2006), (2007), and construct topological groups with very strong combinatorial properties.


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  • 1. T. Banakh, D. Repovš, and L. Zdomskyy, $ o$-Boundedness of free topological groups, Topology Appl. 157 (2010), 466-481. MR 2563296 (2010j:54058)
  • 2. T. Bartoszyński and H. Judah, Set Theory: On the structure of the real line, A. K. Peters, Massachusetts: 1995. MR 1350295 (96k:03002)
  • 3. T. Bartoszyński and B. Tsaban, Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures, Proceedings of the American Mathematical Society 134 (2006), 605-615. MR 2176030 (2006f:54038)
  • 4. A. 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. http://www.math.lsa.umich.edu/˜ablass/hbk.pdf
  • 5. J. Brendle, Generic constructions of small sets of reals, Topology and its Applications 71 (1996), 125-147. MR 1399552 (97i:03030)
  • 6. L. Bukovský and J. Haleš, On Hurewicz properties, Topology and its Applications 132 (2003), 71-79. MR 1990080 (2004f:03085)
  • 7. L. Bukovský, I. Recław, and M. Repický, Spaces not distinguishing pointwise and quasinormal convergence of real functions, Topology and its Applications 41 (1991), 25-41. MR 1129696 (93b:54037)
  • 8. L. Bukovský, I. Recław, and M. Repický, Spaces not distinguishing convergences of real-valued functions, Topology and its Applications 112 (2001), 13-40. MR 1815270 (2002e:54010)
  • 9. K. Ciesielski, A. Millán, and J. Pawlikowski, Uncountable $ \gamma$-sets under axiom $ \mathrm{CPA}^\mathrm{game} \mathrm{cube}$, Fundamenta Mathematicae 176 (2003), 143-155. MR 1971305 (2004b:03077)
  • 10. M. Džamonja, M. Hruśak, and J. Moore, Parametrized $ \lozenge$ principles, Transactions of the American Mathematical Society 356 (2004), 2281-2306. MR 2048518 (2005b:03114)
  • 11. D. Fremlin and A. Miller, On some properties of Hurewicz, Menger and Rothberger, Fundamenta Mathematica 129 (1988), 17-33. MR 954892 (89g:54061)
  • 12. F. Galvin and A. Miller, $ \gamma$-sets and other singular sets of real numbers, Topology and its Applications 17 (1984), 145-155. MR 738943 (85f:54011)
  • 13. J. Gerlits and Zs. Nagy, Some properties of $ C(X)$. I, Topology and its Applications 14 (1982), 151-161. MR 667661 (84f:54021)
  • 14. Gary Gruenhage and Paul Szeptycki, Fréchet Urysohn for finite sets, Topology and its Applications 151 (2005) 238-259. MR 2139755 (2006i:54004)
  • 15. F. Jordan, There are no hereditary productive $ \gamma$-spaces, Topology and its Applications 155 (2008), 1786-1791. MR 2445301 (2009f:54029)
  • 16. H. Judah, S. Shelah, and H. Woodin, The Borel conjecture, Annals of Pure and Applied Logic 50 (1990), 255-269. MR 1086456 (91m:03052)
  • 17. 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)
  • 18. Lj. Kočinac and M. Scheepers, Combinatorics of open covers. VII. Groupability, Fundamenta Mathematicae 179 (2003), 131-155. MR 2029229 (2005c:54020)
  • 19. Lj. Kočinac, 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)
  • 20. A. Miller, A nonhereditary Borel-cover $ \gamma$-set, Real Analysis Exchange 29 (2003/4), 601-606. MR 2083799 (2005g:03079)
  • 21. A. Miller, The $ \gamma$-Borel conjecture, Archive for Mathematical Logic 44 (2005), 425-434. Appendix in http://arxiv.org/math/math.LO/0312308. MR 2139770 (2005m:03096)
  • 22. A. Miller, Cardinal characteristic for relative $ \gamma$-sets, Topology and its Applications 156 (2009), 872-878. MR 2498919
  • 23. M. Sakai, Property $ C''$ and function spaces, Proceedings of the American Mathematical Society 104 (1988), 917-919. MR 964873 (89m:54004)
  • 24. M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology and its Applications 69 (1996), 31-62. MR 1378387 (97h:90123)
  • 25. M. Scheepers, $ C_p(X)$ and Arhangel'skiĭ's $ \alpha_i$ spaces, Topology and its Applications 89 (1998), 265-275. MR 1645184 (99g:54018)
  • 26. M. Scheepers, Sequential convergence in $ {\sf C}_p(X)$ and a covering property, East-West Journal of Mathematics 1 (1999), 207-214. MR 1727383 (2000i:54014)
  • 27. M. Scheepers, Selection principles and covering properties in topology, Note di Matematica 22 (2003), 3-41. MR 2112729 (2005h:54024)
  • 28. M. Scheepers and B. Tsaban, The combinatorics of Borel covers, Topology and its Applications 121 (2002), 357-382. MR 1908999 (2003e:03091)
  • 29. M. Tkachenko, Topological groups for topologists. II, Boletín de la Sociedad Matemática Mexicana. Tercera Serie 6 (2000), 1-41. MR 1768506 (2001g:22004)
  • 30. S. Todorčević, Aronszajn orderings, Duro Kurepa memorial volume, Publications de l'Institut Mathématique (new series) 57 (1995), 29-46. MR 1387352 (98g:03116)
  • 31. B. Tsaban, Some new directions in infinite-combinatorial topology, Set Theory, J. Bagaria and S. Todorčevic, eds., Trends in Mathematics, Birkhäuser (2006), 225-255. MR 2267150 (2007f:03064)
  • 32. 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)
  • 33. B. Tsaban, Additivity numbers of covering properties, Selection Principles and Covering Properties in Topology (L. Kočinac, ed.), Quaderni di Matematica 18, Seconda Universita di Napoli, Caserta (2006), 245-282. MR 2395757 (2009e:03088)
  • 34. B. Tsaban, A new selection principle, Topology Proceedings 31 (2007), 319-329. MR 2363173 (2008j:54017)
  • 35. B. Tsaban, Selection principles and special sets of reals, Open Problems in Topology II, E. Pearl, ed., Elsevier B.V. (2007), 91-108.
  • 36. B. Tsaban and L. Zdomskyy, Combinatorial images of sets of reals and semifilter trichotomy, Journal of Symbolic Logic, 73 (2008), 1278-1288. MR 2467216 (2009h:03064)
  • 37. B. Tsaban and L. Zdomskyy, Hereditarily Hurewicz spaces and Arhangel'skiĭ sheaf amalgamations, Journal of the European Mathematical Society, to appear.

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Additional Information

Tal Orenshtein
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Email: talo@weizmann.ac.il

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

DOI: https://doi.org/10.1090/S0002-9947-2011-05228-1
Received by editor(s): August 2, 2009
Received by editor(s) in revised form: October 20, 2009
Published electronically: February 7, 2011
Article copyright: © Copyright 2011 American Mathematical Society

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