Linear 𝜎-additivity and some applications

Orenshtein, Tal; Tsaban, Boaz

doi:10.1090/S0002-9947-2011-05228-1

Linear $\sigma$-additivity and some applications
HTML articles powered by AMS MathViewer

by Tal Orenshtein and Boaz Tsaban PDF

Trans. Amer. Math. Soc. 363 (2011), 3621-3637 Request permission

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.

References

Taras Banakh, Dušan Repovš, and Lyubomyr Zdomskyy, $o$-boundedness of free topological groups, Topology Appl. 157 (2010), no. 2, 466–481. MR 2563296, DOI 10.1016/j.topol.2009.10.006
Tomek Bartoszyński and Haim Judah, Set theory, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR 1350295
Tomek Bartoszyński and Boaz Tsaban, Hereditary topological diagonalizations and the Menger-Hurewicz conjectures, Proc. Amer. Math. Soc. 134 (2006), no. 2, 605–615. MR 2176030, DOI 10.1090/S0002-9939-05-07997-9
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
Jörg Brendle, Generic constructions of small sets of reals, Topology Appl. 71 (1996), no. 2, 125–147. MR 1399552, DOI 10.1016/0166-8641(95)00071-2
Lev Bukovský and Jozef Haleš, On Hurewicz properties, Topology Appl. 132 (2003), no. 1, 71–79. MR 1990080, DOI 10.1016/S0166-8641(02)00364-4
Lev Bukovský, Ireneusz Recław, and Miroslav Repický, Spaces not distinguishing pointwise and quasinormal convergence of real functions, Topology Appl. 41 (1991), no. 1-2, 25–40. MR 1129696, DOI 10.1016/0166-8641(91)90098-7
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, DOI 10.1016/S0166-8641(99)00226-6
Krzysztof Ciesielski, Andrés Millán, and Janusz Pawlikowski, Uncountable $\gamma$-sets under axiom $\rm CPA^{game}_{cube}$, Fund. Math. 176 (2003), no. 2, 143–155. MR 1971305, DOI 10.4064/fm176-2-3
Justin Tatch Moore, Michael Hrušák, and Mirna D amonja, Parametrized $\diamondsuit$ principles, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2281–2306. MR 2048518, DOI 10.1090/S0002-9947-03-03446-9
Arnold W. Miller and David H. Fremlin, On some properties of Hurewicz, Menger, and Rothberger, Fund. Math. 129 (1988), no. 1, 17–33. MR 954892, DOI 10.4064/fm-129-1-17-33
Fred Galvin and Arnold W. Miller, $\gamma$-sets and other singular sets of real numbers, Topology Appl. 17 (1984), no. 2, 145–155. MR 738943, DOI 10.1016/0166-8641(84)90038-5
J. Gerlits and Zs. Nagy, Some properties of $C(X)$. I, Topology Appl. 14 (1982), no. 2, 151–161. MR 667661, DOI 10.1016/0166-8641(82)90065-7
Gary Gruenhage and Paul J. Szeptycki, Fréchet-Urysohn for finite sets, Topology Appl. 151 (2005), no. 1-3, 238–259. MR 2139755, DOI 10.1016/j.topol.2003.09.014
Francis Jordan, There are no hereditary productive $\gamma$-spaces, Topology Appl. 155 (2008), no. 16, 1786–1791. MR 2445301, DOI 10.1016/j.topol.2008.05.016
Haim Judah, Saharon Shelah, and W. H. Woodin, The Borel conjecture, Ann. Pure Appl. Logic 50 (1990), no. 3, 255–269. MR 1086456, DOI 10.1016/0168-0072(90)90058-A
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 10.1016/S0166-8641(96)00075-2
Ljubiša D. R. Kočinac and Marion Scheepers, Combinatorics of open covers. VII. Groupability, Fund. Math. 179 (2003), no. 2, 131–155. MR 2029229, DOI 10.4064/fm179-2-2
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
Arnold W. Miller, A nonhereditary Borel-cover $\gamma$-set, Real Anal. Exchange 29 (2003/04), no. 2, 601–606. MR 2083799, DOI 10.14321/realanalexch.29.2.0601
Arnold W. Miller, The $\gamma$-Borel conjecture, Arch. Math. Logic 44 (2005), no. 4, 425–434. MR 2139770, DOI 10.1007/s00153-004-0260-0
Arnold W. Miller, The cardinal characteristic for relative $\gamma$-sets, Topology Appl. 156 (2009), no. 5, 872–878. MR 2498919, DOI 10.1016/j.topol.2008.11.006
Masami Sakai, Property C′′and function spaces, Proc. Amer. Math. Soc. 104 (1988), no. 3, 917–919. MR 964873, DOI 10.1090/S0002-9939-1988-0964873-0
Marion Scheepers, Combinatorics of open covers. I. Ramsey theory, Topology Appl. 69 (1996), no. 1, 31–62. MR 1378387, DOI 10.1016/0166-8641(95)00067-4
Marion Scheepers, $C_p(X)$ and Arhangel′skiĭ’s $\alpha _i$-spaces, Topology Appl. 89 (1998), no. 3, 265–275. MR 1645184, DOI 10.1016/S0166-8641(97)00218-6
Marion Scheepers, Sequential convergence in $C_p(X)$ and a covering property, East-West J. Math. 1 (1999), no. 2, 207–214. MR 1727383
Marion Scheepers, Selection principles and covering properties in topology, Note Mat. 22 (2003/04), no. 2, 3–41. MR 2112729
Marion Scheepers and Boaz Tsaban, The combinatorics of Borel covers, Topology Appl. 121 (2002), no. 3, 357–382. MR 1908999, DOI 10.1016/S0166-8641(01)00078-5
Mikhail G. Tkachenko, Topological groups for topologists. II, Bol. Soc. Mat. Mexicana (3) 6 (2000), no. 1, 1–41. MR 1768506
S. Todorčević, Aronszajn orderings, Publ. Inst. Math. (Beograd) (N.S.) 57(71) (1995), 29–46. Đuro Kurepa memorial volume. MR 1387352
Boaz Tsaban, Some new directions in infinite-combinatorial topology, Set theory, Trends Math., Birkhäuser, Basel, 2006, pp. 225–255. MR 2267150, DOI 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 10.1090/S0002-9939-05-08034-2
Boaz Tsaban, Additivity numbers of covering properties, Selection principles and covering properties in topology, Quad. Mat., vol. 18, Dept. Math., Seconda Univ. Napoli, Caserta, 2006, pp. 245–282. MR 2395757
Boaz Tsaban, A new selection principle, Topology Proc. 31 (2007), no. 1, 319–329. MR 2363173
B. Tsaban, Selection principles and special sets of reals, Open Problems in Topology II, E. Pearl, ed., Elsevier B.V. (2007), 91–108.
Boaz Tsaban and Lyubomyr Zdomskyy, Combinatorial images of sets of reals and semifilter trichotomy, J. Symbolic Logic 73 (2008), no. 4, 1278–1288. MR 2467216, DOI 10.2178/jsl/1230396918
B. Tsaban and L. Zdomskyy, Hereditarily Hurewicz spaces and Arhangel’skiĭ sheaf amalgamations, Journal of the European Mathematical Society, to appear.