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Special subsets of the reals and tree forcing notions

Authors: Marcin Kysiak, Andrzej Nowik and Tomasz Weiss
Journal: Proc. Amer. Math. Soc. 135 (2007), 2975-2982
MSC (2000): Primary 03E05, 03E35, 28E15, 54G99
Published electronically: May 9, 2007
MathSciNet review: 2317976
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Abstract: We study relationships between classes of special subsets of the reals (e.g. meager-additive sets, $ \gamma$-sets, $ C''$-sets, $ \lambda$-sets) and the ideals related to the forcing notions of Laver, Mathias, Miller and Silver.

References [Enhancements On Off] (What's this?)

  • 1. T. Bartoszynski and H. Judah, Set Theory: On the Structure of the Real Line, A K Peters, Ltd, Wellesley, Massachusetts, 1995. MR 1350295 (96k:03002)
  • 2. J. Brendle and B. Löwe, Solovay-type characterizations for forcing-algebras, J. Symbolic Logic 64 (1999), no. 3, 1307-1323. MR 1779764 (2001h:03086)
  • 3. J. B. Brown, The Ramsey sets and related sigma algebras and ideals, Fundamenta Mathematicae 136 (1990), no. 3, 179-185. MR 1095690 (92a:04003)
  • 4. L. Halbeisen, Making doughnuts of Cohen reals, Mathematical Logic Quarterly 49 (2003), no. 2, 173-178. MR 1961459 (2004b:03080)
  • 5. M. Kysiak and T. Weiss, Small subsets of the reals and tree forcing notions, Proceedings of the American Mathematical Society 132 (2004), no. 1, 251-259.MR 2021269 (2004j:03062)
  • 6. A. W. Miller, Handbook of set-theoretic topology, ch. Special Subsets of the Real Line, pp. 201-233, North-Holland, Amsterdam, 1984.
  • 7. -, Special sets of reals, Israel Mathematical Conference Proceedings, vol. 6, 1993. MR 1234286 (95a:03064)
  • 8. -, A hodgepodge of sets of reals, Preprint (2006).
  • 9. A. Nowik, M. Scheepers, and T. Weiss, The algebraic sums of sets of real numbers with strong measure zero sets, Journal of Symbolic Logic 63 (1998), no. 1, 301-324.MR 1610427 (99c:54049)
  • 10. A. Nowik and T. Weiss, On the Ramseyan properties of some special subsets of $ 2^{\omega}$ and their algebraic sums, The Journal of Symbolic Logic 67 (2002), no. 2, 547-556.MR 1905154 (2003c:03087)
  • 11. -, Strongly meager sets of the real numbers and tree forcing notions, Proceedings of the American Mathematical Society 130 (2002), no. 4, 1183-1187.MR 1873795 (2002j:03049)
  • 12. J. C. Oxtoby, Measure and category, Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York - Berlin, 1980. MR 0584443 (81j:28003)

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Additional Information

Marcin Kysiak
Affiliation: Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland

Andrzej Nowik
Affiliation: Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

Tomasz Weiss
Affiliation: Institute of Mathematics and Physics, Akademia Podlaska, ul. 3-go Maja, 08-110 Siedlce, Poland

Keywords: $\gamma$-set, Rothberger's property, meager-additive set, $\sigma$-set, Laver forcing, Miller forcing, Silver forcing, completely Ramsey-null set
Received by editor(s): March 13, 2006
Received by editor(s) in revised form: June 8, 2006
Published electronically: May 9, 2007
Additional Notes: A part of the research was made when the first author was visiting the Institute of Mathematics of the Polish Academy of Sciences
The second author was partially supported by grant BW/5100-5-0201-6
Communicated by: Julia Knight
Article copyright: © Copyright 2007 American Mathematical Society

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