Archive for Mathematical Logic

, Volume 52, Issue 7–8, pp 827–846 | Cite as

Strongly dominating sets of reals

Article

Abstract

We analyze the structure of strongly dominating sets of reals introduced in Goldstern et al. (Proc Am Math Soc 123(5):1573–1581, 1995). We prove that for every \({\kappa < \mathfrak{b}}\) a \({\kappa}\) -Suslin set \({A\subseteq{}^\omega\omega}\) is strongly dominating if and only if A has a Laver perfect subset. We also investigate the structure of the class l of Baire sets for the Laver category base and compare the σ-ideal of sets which are not strongly dominating with the Laver ideal l0.

Keywords

Strongly dominating sets Laver perfect sets \({\kappa}\) -Suslin sets Laver category base Domination game 

Mathematics Subject Classification (2000)

Primary 03E15 Secondary 03E17 03E50 91A44 

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References

  1. 1.
    Bartoszyński T., Judah H.: Set Theory: On the Structure of the Real Line. A K Peters Ltd., Welleslay, Massachusetts (1995)MATHGoogle Scholar
  2. 2.
    Brendle J., Hjorth G., Spinas O.: Regularity properties for dominating projective sets. Ann. Pure Appl. Log. 72, 291–307 (1995)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Goldstern M., Repický M., Shelah S., Spinas O.: On tree ideals. Proc. Am. Math. Soc. 123(5), 1573–1581 (1995)CrossRefMATHGoogle Scholar
  4. 4.
    Judah H., Miller A., Shelah S.: Sacks forcing, Laver forcing and Martin’s axiom. Arch. Math. Log. 31, 145–161 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kechris A.S.: On a notion of smallness for subsets of the Baire space. Trans. Am. Math. Soc. 229, 191–207 (1977)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kułaga W.: On fields and ideals connected with notions of forcing. Coll. Math. 105(2), 271–281 (2006)CrossRefMATHGoogle Scholar
  7. 7.
    Miller, A.W.: Hechler and Laver Trees, preprint. http://arxiv.org/abs/1204.5198
  8. 8.
    Morgan J.C. II: Point Set Theory. Dekker, New York (1990)MATHGoogle Scholar
  9. 9.
    Moschovakis, Y.N.: Descriptive Set Theory, Mathematical Surveys and Monographs, 2nd edn., vol. 155. AMS, Providence, RI (2009)Google Scholar
  10. 10.
    Repický, M.: Bases of measurability in Boolean algebras. Math. Slovaca, to appearGoogle Scholar
  11. 11.
    Spinas O.: Dominating projective sets in the Baire space. Ann. Pure Appl. Log. 68, 327–342 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    van Douwen E.K. : The integers and topology, chapter 3. In: Kunen, K., Vaughan, J.E. (eds) Handbook of Set-Theoretic Topology, pp. 111–167. North-Holland, Amsterdam (1984)Google Scholar
  13. 13.
    Zapletal J.: Isolating cardinal invariants. J. Math. Log. 3(1), 143–162 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute of MathematicsP. J. Šafárik UniversityKošiceSlovak Republic
  2. 2.Institute of MathematicsSlovak Academy of SciencesKošiceSlovak Republic

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