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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Since we shall not analyze this poset directly but rather use certain topological characterizations, we refer the reader to, e.g., [2] for its definition.
This set is infinite by the definition of F and \(f_\alpha \in F\).
We believe that this straightforward argument is well-known, but we were unable to locate it in the literature.
References
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)
Brendle, J.: Mob families and mad families. Arch. Math. Logic 37, 183–197 (1998)
Chaber, J., Pol, R.: A remark on Fremlin–Miller theorem concerning the Menger property and Michael concentrated sets, Preprint (2002)
Chodounský, D., Repovš, D., Zdomskyy, L.: Mathias forcing and combinatorial covering properties of filters. J. Symb. Log. 80, 1398–1410 (2015)
Guzmán, O., Hrušák, M., Martínez, A.A.: Canjar filters. Notre Dame J. Formal Log. 58, 79–95 (2017)
Hurewicz, W.: Über die Verallgemeinerung des Borellschen Theorems. Math. Z. 24, 401–421 (1925)
Hurewicz, W.: Über Folgen stetiger Funktionen. Fund. Math. 9, 193–204 (1927)
Just, W., Miller, A.W., Scheepers, M., Szeptycki, P.J.: The combinatorics of open covers II. Topol. Appl. 73, 241–266 (1996)
Kechris, A.: Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156. Springer, Berlin (1995)
Kurilić, M.S.: Cohen-stable families of subsets of integers. J. Symb. Log. 66, 257–270 (2001)
Laver, R.: On the consistency of Borel’s conjecture. Acta Math. 137, 151–169 (1976)
Malykhin, V.I.: Topological properties of Cohen generic extensions. Trans. Moscow Math. Soc. 52, 1–32 (1990)
Menger, K.: Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte. Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie) 133, 421–444 (1924)
Repovš, D., Zdomskyy, L.: Products of Hurewicz spaces in the Laver model. Bull. Symb. Log. 23, 324–333 (2017)
Tsaban, B., Zdomskyy, L.: Scales, fields, and a problem of Hurewicz. J. Eur. Math. Soc. 10(3), 837–866 (2008)
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 \( cov ({\mathcal {N}})=\mathfrak c\).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author was supported by FAPESP (2017/09252-3). The second author would like to thank the Austrian Science Fund FWF (Grant I 2374-N35) for generous support for this research.
Rights and permissions
About this article
Cite this article
Aurichi, L., Zdomskyy, L. Covering properties of \(\omega \)-mad families. Arch. Math. Logic 59, 445–452 (2020). https://doi.org/10.1007/s00153-019-00700-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-019-00700-y