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Point-cofinite covers in the Laver model. (English) Zbl 1200.03038

Summary: Let \({S}_1(\Gamma ,\Gamma )\) be the statement: For each sequence of point-cofinite open covers, one can pick one element from each cover and obtain a point-cofinite cover. \(\mathfrak{b}\) is the minimal cardinality of a set of reals not satisfying \({S}_1(\Gamma ,\Gamma )\). We prove the following assertions: 6mm
(1)
If there is an unbounded tower, then there are sets of reals of cardinality \(\mathfrak{b}\) satisfying \({S}_1(\Gamma ,\Gamma )\).
(2)
It is consistent that all sets of reals satisfying \({S}_1(\Gamma ,\Gamma )\) have cardinality smaller than \(\mathfrak{b}\).
These results can also be formulated as dealing with Arkhangel’skiĭ’s property \(\alpha_2\) for spaces of continuous real-valued functions.
The main technical result is that in Laver’s model, each set of reals of cardinality \(\mathfrak{b}\) has an unbounded Borel image in the Baire space \(\omega ^{\omega }\).

MSC:

03E35 Consistency and independence results
03E17 Cardinal characteristics of the continuum
26A03 Foundations: limits and generalizations, elementary topology of the line
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[1] Tomek Bartoszynski and Saharon Shelah, Continuous images of sets of reals, Topology Appl. 116 (2001), no. 2, 243 – 253. · Zbl 0992.03061
[2] Tomek Bartoszyński and Boaz Tsaban, Hereditary topological diagonalizations and the Menger-Hurewicz conjectures, Proc. Amer. Math. Soc. 134 (2006), no. 2, 605 – 615. · Zbl 1137.54018
[3] James E. Baumgartner and Richard Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), no. 3, 271 – 288. · Zbl 0427.03043
[4] A. Blass, Combinatorial cardinal characteristics of the continuum, in: Handbook of Set Theory , Kluwer Academic Publishers, Dordrecht, to appear. http://www.math.lsa.umich.edu/ãblass/hbk.pdf · Zbl 1198.03058
[5] Alan Dow, Two classes of Fréchet-Urysohn spaces, Proc. Amer. Math. Soc. 108 (1990), no. 1, 241 – 247. · Zbl 0675.54029
[6] Fred Galvin and Arnold W. Miller, \?-sets and other singular sets of real numbers, Topology Appl. 17 (1984), no. 2, 145 – 155. · Zbl 0551.54001
[7] J. Gerlits and Zs. Nagy, Some properties of \?(\?). I, Topology Appl. 14 (1982), no. 2, 151 – 161. · Zbl 0503.54020
[8] W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Mathematische Zeitschrift 24 (1925), 401-421. · JFM 51.0454.02
[9] 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. , https://doi.org/10.1016/S0166-8641(96)00075-2 Marion Scheepers, Combinatorics of open covers. III. Games, \?_{\?}(\?), Fund. Math. 152 (1997), no. 3, 231 – 254. · Zbl 0870.03021
[10] Ljubiša D. R. Kočinac, Selection principles related to \?\?-properties, Taiwanese J. Math. 12 (2008), no. 3, 561 – 571. · Zbl 1153.54009
[11] Richard Laver, On the consistency of Borel’s conjecture, Acta Math. 137 (1976), no. 3-4, 151 – 169. · Zbl 0357.28003
[12] Arnold W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), no. 3, 575 – 584. · Zbl 0527.03031
[13] Peter J. Nyikos, Subsets of ^{\?}\? and the Fréchet-Urysohn and \?\?-properties, Topology Appl. 48 (1992), no. 2, 91 – 116. · Zbl 0774.54019
[14] Masami Sakai, The sequence selection properties of \?_{\?}(\?), Topology Appl. 154 (2007), no. 3, 552 – 560. · Zbl 1109.54014
[15] Marion Scheepers, Combinatorics of open covers. I. Ramsey theory, Topology Appl. 69 (1996), no. 1, 31 – 62. · Zbl 0848.54018
[16] Marion Scheepers, \?_{\?}(\?) and Arhangel\(^{\prime}\)skiĭ’s \?\?-spaces, Topology Appl. 89 (1998), no. 3, 265 – 275. · Zbl 0930.54017
[17] Marion Scheepers, Sequential convergence in \?_{\?}(\?) and a covering property, East-West J. Math. 1 (1999), no. 2, 207 – 214. · Zbl 0976.54016
[18] Marion Scheepers and Boaz Tsaban, The combinatorics of Borel covers, Topology Appl. 121 (2002), no. 3, 357 – 382. · Zbl 1025.03042
[19] B. Tsaban, Menger’s and Hurewicz’s Problems: Solutions from “The Book” and refinements, Contemporary Mathematics, American Mathematical Society, to appear. · Zbl 1252.03111
[20] B. Tsaban and L. Zdomskyy, Hereditarily Hurewicz spaces and Arhangel’skiĭ sheaf amalgamations, Journal of the European Mathematical Society, to appear. · Zbl 1267.54019
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