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Covering properties of \(\omega \)-mad families

  • Leandro Aurichi
  • Lyubomyr ZdomskyyEmail author
Article

Abstract

We prove that Martin’s Axiom implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while \(\mathfrak b=\mathfrak c\) is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel’s conjecture.

Keywords

Menger space Mad family Cohen forcing Laver forcing 

Mathematics Subject Classification

Primary 03E35 54D20 Secondary 03E05 

Notes

Acknowledgements

The work reported here was carried out during the visit of the second named author at the Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, in July 2018. This visit was supported by FAPESP (2017/09252-3). The second named author thanks the first named author and Lucia Junqueira for their kind hospitality. In addition, we thank Osvaldo Guzman for his comments on the previous version of this paper. We are also grateful to the anonymous referee who among other things pointed out that our proof of Theorem 1.1 requires also the assumption \( cov ({\mathcal {N}})=\mathfrak c\).

References

  1. 1.
    Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A., Magidor, M. (eds.) Handbook of Set Theory, pp. 395–491. Springer, Berlin (2010)CrossRefGoogle Scholar
  2. 2.
    Brendle, J.: Mob families and mad families. Arch. Math. Logic 37, 183–197 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chaber, J., Pol, R.: A remark on Fremlin–Miller theorem concerning the Menger property and Michael concentrated sets, Preprint (2002)Google Scholar
  4. 4.
    Chodounský, D., Repovš, D., Zdomskyy, L.: Mathias forcing and combinatorial covering properties of filters. J. Symb. Log. 80, 1398–1410 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Guzmán, O., Hrušák, M., Martínez, A.A.: Canjar filters. Notre Dame J. Formal Log. 58, 79–95 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hurewicz, W.: Über die Verallgemeinerung des Borellschen Theorems. Math. Z. 24, 401–421 (1925)CrossRefGoogle Scholar
  7. 7.
    Hurewicz, W.: Über Folgen stetiger Funktionen. Fund. Math. 9, 193–204 (1927)CrossRefGoogle Scholar
  8. 8.
    Just, W., Miller, A.W., Scheepers, M., Szeptycki, P.J.: The combinatorics of open covers II. Topol. Appl. 73, 241–266 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kechris, A.: Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156. Springer, Berlin (1995)CrossRefGoogle Scholar
  10. 10.
    Kurilić, M.S.: Cohen-stable families of subsets of integers. J. Symb. Log. 66, 257–270 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Laver, R.: On the consistency of Borel’s conjecture. Acta Math. 137, 151–169 (1976)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Malykhin, V.I.: Topological properties of Cohen generic extensions. Trans. Moscow Math. Soc. 52, 1–32 (1990)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Menger, K.: Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte. Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie) 133, 421–444 (1924)Google Scholar
  14. 14.
    Repovš, D., Zdomskyy, L.: Products of Hurewicz spaces in the Laver model. Bull. Symb. Log. 23, 324–333 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Tsaban, B., Zdomskyy, L.: Scales, fields, and a problem of Hurewicz. J. Eur. Math. Soc. 10(3), 837–866 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São Paulo (ICMC-USP)São CarlosBrazil
  2. 2.Institut für Diskrete Mathematik und GeometrieTechnische Universität WienViennaAustria

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