Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-16T07:27:32.020Z Has data issue: false hasContentIssue false
  • English
  • Français

PRODUCTS OF HUREWICZ SPACES IN THE LAVER MODEL

Published online by Cambridge University Press:  04 December 2017

DUŠAN REPOVŠ  and
LYUBOMYR ZDOMSKYY
DUŠAN REPOVŠ
Affiliation:
FACULTY OF EDUCATION, AND FACULTY OF MATHEMATICS AND PHYSICS UNIVERSITY OF LJUBLJANA LJUBLJANA1000, SLOVENIAE-mail: [email protected]: http://www.fmf.uni-lj.si/∼repovs/index.htm
LYUBOMYR ZDOMSKYY
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITY OF VIENNA, WÄHRINGER STRASSE 25 A-1090 WIEN, AUSTRIAE-mail: [email protected]: http://www.logic.univie.ac.at/∼lzdomsky/
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Primary 03E3554D20Secondary 54C5003E05Menger spaceHurewicz spacesemifilterLaver forcing
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 23 , Issue 3 , September 2017 , pp. 324 - 333
Copyright
Copyright © The Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aurichi, L. F., D-spaces, topological games, and selection principles . Topology Proceedings, vol. 36 (2010), pp. 107–122.Google Scholar
Babinkostova, L., On some questions about selective separability . Mathematical Logic Quarterly, 55 (2009), pp. 539–541.Google Scholar
Barman, D. and Dow, A., Proper forcing axiom and selective separability . Topology and its Applications, vol. 159 (2012), pp. 806–813.CrossRefGoogle Scholar
Bartoszyński, T. and Judah, H., Set Theory. On the Structure of the Real Line, A. K. Peters, Ltd., Wellesley, MA, 1995.Google Scholar
Blass, A., Combinatorial cardinal characteristics of the continuum, Handbook of Set Theory (Foreman, M., Kanamori, A., and Magidor, M., editors), Springer, Dordrecht, 2010, pp. 395–491.Google Scholar
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. 213–243.CrossRefGoogle Scholar
Bukovsky, L., The Structure of the Real Line, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series), 71, Birkhäuser/Springer Basel AG, Basel, 2011.Google Scholar
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. 493–504.Google Scholar
Chodounský, D., Repovš, D., and Zdomskyy, L., Mathias forcing and combinatorial covering properties of filters . The Journal of Symbolic Logic, vol. 80 (2015), pp. 1398–1410.CrossRefGoogle Scholar
Gartside, P., Medini, A., and Zdomskyy, L., The Tukey Order, Hyperspaces, and Selection Principles, work in progress.Google Scholar
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.Google Scholar
Hurewicz, W., Über die Verallgemeinerung des Borelschen Theorems . Mathematische Zeitschrift, vol. 24 (1925), pp. 401–421.CrossRefGoogle Scholar
Hurewicz, W., Über Folgen stetiger Funktionen . Fundamenta Mathematicae, vol. 9 (1927), pp. 193–204.CrossRefGoogle Scholar
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. 241–266.Google Scholar
Laver, R., On the consistency of Borel’s conjecture . Acta Mathematica, vol. 137 (1976), pp. 151–169.CrossRefGoogle Scholar
Menger, K., Einige Überdeckungssätze der Punktmengenlehre . Sitzungsberichte. Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie), vol. 133 (1924), pp. 421–444.Google Scholar
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. 143–159.CrossRefGoogle Scholar
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. 201–233.CrossRefGoogle Scholar
Miller, A. W. and Tsaban, B., Point-cofinite covers in the Laver model . Proceedings of the American Mathematical Society, vol. 138 (2010), pp. 3313–3321.Google Scholar
Miller, A. W. and Tsaban, B., Selective covering properties of product spaces . Annals of Pure and Applied Logic, vol. 165 (2014), pp. 1034–1057.CrossRefGoogle Scholar
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. 2865–2889.Google Scholar
Repovš, D. and Zdomskyy, L., On M-separability of countable spaces and function spaces . Topology and its Applications, vol. 157 (2010), pp. 2538–2541.Google Scholar
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. 331–355.CrossRefGoogle Scholar
Scheepers, M., Combinatorics of open covers. I. Ramsey theory . Topology and its Applications, vol. 69 (1996), pp. 31–62.Google Scholar
Scheepers, M. and Tall, F., Lindelöf indestructibility, topological games and selection principles . Fundamenta Mathematicae, vol. 210 (2010), pp. 1–46.Google Scholar
Sierpiński, W., Sur un ensemble non dénombrable, dont toute image continue est de mesure nulle . Fundamenta Mathematicae, vol. 11 (1928), pp. 302–303.CrossRefGoogle Scholar
Scheepers, M. and Tsaban, B., The combinatorics of Borel covers . Topology and its Applications, vol. 121 (2002), pp. 357–382.Google Scholar
Todorčević, S., Aronszajn orderings. Djuro Kurepa memorial volume . Publications de l’Institut Mathématique (Beograd), vol. 57 (1995), no. 71, pp. 29–46.Google Scholar
Tsaban, B., Selection principles and special sets of reals, Open Problems in Topology II (Pearl, E., editor), Elsevier Science Publishing, Amsterdam, 2007, pp. 91–108.Google Scholar
Tsaban, B., Algebra, selections, and additive Ramsey theory, preprint, 2015, http://arxiv.org/pdf/1407.7437.pdf.Google Scholar
Tsaban, B. and Zdomskyy, L., Hereditarily Hurewicz spaces and Arhangel’ski sheaf amalgamations . Journal of the European Mathematical Society, vol. 14 (2012), pp. 353–372.Google Scholar
Zdomskyy, L., A semifilter approach to selection principles . Commentationes Mathematicae Universitatis Carolinae, vol. 46 (2005), pp. 525–539.Google Scholar
Zdomskyy, L., Products of Menger spaces in the Miller model, preprint, http://www.logic.univie.ac.at/∼lzdomsky/.Google Scholar