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PRODUCTS OF HUREWICZ SPACES IN THE LAVER MODEL

  • DUŠAN REPOVŠ (a1) and LYUBOMYR ZDOMSKYY (a2)
Abstract
Abstract

This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel’s conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.

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[1] Aurichi L. F., D-spaces, topological games, and selection principles . Topology Proceedings, vol. 36 (2010), pp. 107122.
[2] Babinkostova L., On some questions about selective separability . Mathematical Logic Quarterly, 55 (2009), pp. 539541.
[3] Barman D. and Dow A., Proper forcing axiom and selective separability . Topology and its Applications, vol. 159 (2012), pp. 806813.
[4] Bartoszyński T. and Judah H., Set Theory. On the Structure of the Real Line, A. K. Peters, Ltd., Wellesley, MA, 1995.
[5] Blass A., Combinatorial cardinal characteristics of the continuum, Handbook of Set Theory (Foreman M., Kanamori A., and Magidor M., editors), Springer, Dordrecht, 2010, pp. 395491.
[6] Blass A. and Shelah S., There may be simple ${P_{{\aleph _1}}}$ - and ${P_{{\aleph _2}}}$ -points and the Rudin-Keisler ordering may be downward directed . Annals of Pure and Applied Logic, vol. 33 (1987), pp. 213243.
[7] Bukovsky L., The Structure of the Real Line, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series), 71, Birkhäuser/Springer Basel AG, Basel, 2011.
[8] Chodounsky D., Guzman O., and Hrušák M., Mathias-Prikry and Laver type forcing; Summable ideals, coideals, and +-selective filters . Archive for Mathematical Logic, vol. 55 (2016), pp. 493504.
[9] Chodounský D., Repovš D., and Zdomskyy L., Mathias forcing and combinatorial covering properties of filters . The Journal of Symbolic Logic, vol. 80 (2015), pp. 13981410.
[10] Gartside P., Medini A., and Zdomskyy L., The Tukey Order, Hyperspaces, and Selection Principles, work in progress.
[11] Hrušák M. and van Mill J., The existence of a meager in itself CDH metric space is independent of ZFC . Proceedings of the American Mathematical Society, doi:10.1090/proc/13434.
[12] Hurewicz W., Über die Verallgemeinerung des Borelschen Theorems . Mathematische Zeitschrift, vol. 24 (1925), pp. 401421.
[13] Hurewicz W., Über Folgen stetiger Funktionen . Fundamenta Mathematicae, vol. 9 (1927), pp. 193204.
[14] Just W., Miller A. W., Scheepers M., and Szeptycki P. J., The combinatorics of open covers. II . Topology and its Applications, vol. 73 (1996), pp. 241266.
[15] Laver R., On the consistency of Borel’s conjecture . Acta Mathematica, vol. 137 (1976), pp. 151169.
[16] Menger K., Einige Überdeckungssätze der Punktmengenlehre . Sitzungsberichte. Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie), vol. 133 (1924), pp. 421444.
[17] Miller A., Rational perfect set forcing , Axiomatic Set Theory (Baumgartner J., Martin D. A., and Shelah S., editors), Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, RI, 1984, pp. 143159.
[18] Miller A. W., Special subsets of the real line , Handbook of Set-Theoretic Topology (Kunen K. and Vaughan J. E., editors), North Holland, Amsterdam, 1984, pp. 201233.
[19] Miller A. W. and Tsaban B., Point-cofinite covers in the Laver model . Proceedings of the American Mathematical Society, vol. 138 (2010), pp. 33133321.
[20] Miller A. W. and Tsaban B., Selective covering properties of product spaces . Annals of Pure and Applied Logic, vol. 165 (2014), pp. 10341057.
[21] Miller A. W., Tsaban B., and Zdomskyy L., Selective covering properties of product spaces, II: Gamma spaces . Transactions of the American Mathematical Society, vol. 368 (2016), pp. 28652889.
[22] Repovš D. and Zdomskyy L., On M-separability of countable spaces and function spaces . Topology and its Applications, vol. 157 (2010), pp. 25382541.
[23] Sacks G. E., Forcing with perfect closed sets , Axiomatic Set Theory (Scott D., editor), Proceedings of the Symposium on Pure Mathematics, Vol. XIII, Part I, American Mathematical Society, Providence, RI, 1971, pp. 331355.
[24] Scheepers M., Combinatorics of open covers. I. Ramsey theory . Topology and its Applications, vol. 69 (1996), pp. 3162.
[25] Scheepers M. and Tall F., Lindelöf indestructibility, topological games and selection principles . Fundamenta Mathematicae, vol. 210 (2010), pp. 146.
[26] Sierpiński W., Sur un ensemble non dénombrable, dont toute image continue est de mesure nulle . Fundamenta Mathematicae, vol. 11 (1928), pp. 302303.
[27] Scheepers M. and Tsaban B., The combinatorics of Borel covers . Topology and its Applications, vol. 121 (2002), pp. 357382.
[28] Todorčević S., Aronszajn orderings. Djuro Kurepa memorial volume . Publications de l’Institut Mathématique (Beograd), vol. 57 (1995), no. 71, pp. 2946.
[29] Tsaban B., Selection principles and special sets of reals, Open Problems in Topology II (Pearl E., editor), Elsevier Science Publishing, Amsterdam, 2007, pp. 91108.
[30] Tsaban B., Algebra, selections, and additive Ramsey theory, preprint, 2015, http://arxiv.org/pdf/1407.7437.pdf.
[31] Tsaban B. and Zdomskyy L., Hereditarily Hurewicz spaces and Arhangel’ski sheaf amalgamations . Journal of the European Mathematical Society, vol. 14 (2012), pp. 353372.
[32] Zdomskyy L., A semifilter approach to selection principles . Commentationes Mathematicae Universitatis Carolinae, vol. 46 (2005), pp. 525539.
[33] Zdomskyy L., Products of Menger spaces in the Miller model, preprint, http://www.logic.univie.ac.at/∼lzdomsky/.
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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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