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On a rectangle inclusion problem


Author: Janusz Pawlikowski
Journal: Proc. Amer. Math. Soc. 123 (1995), 3189-3195
MSC: Primary 03E05; Secondary 03E15, 54A35
DOI: https://doi.org/10.1090/S0002-9939-1995-1264828-9
MathSciNet review: 1264828
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Abstract: We show that if every set of reals of size $ {2^{{\aleph _0}}}$ contains a meager-to-one continuous image of a set that cannot be covered by less than $ {2^{{\aleph _0}}}$ meager sets, then there exists a null (Lebesgue measure zero) subset of the plane $ \mathbb{R} \times \mathbb{R}$ that meets every nonnull rectangle $ X \times Y$. The antecedent is satisfied, e.g., if $ {\omega _2}$ Cohen reals are added to a model of the continuum hypothesis.


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

  • [1] M. Burke, A theorem of Friedman on rectangle inclusion and its consequences, Note of March 7, 1991.
  • [2] T. Bartoszyński, M. Goldstern, H. Judah, and S. Shelah, All meager filters may be null, Proc. Amer. Math. Soc. 117 (1993), 515-521. MR 1111433 (93d:03055)
  • [3] T. Bartoszyński and H. Judah, Measure and category in set theory (forthcoming book).
  • [4] H. G. Eggleston, Two measure properties of cartesian product sets, Quart. J. Math. Oxford (2) 5 (1954), 108-115. MR 0064850 (16:344e)
  • [5] D. H. Fremlin, Problem list, circulated notes, 1987.
  • [6] D. H. Fremlin and M. Talagrand, A decomposition theorem for additive set functions and applications to Pettis integral and ergodic means, Math. Z. 168 (1979), 117-142. MR 544700 (80k:28004)
  • [7] A. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93-114. MR 613787 (84e:03058a)
  • [8] -, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), 575-584. MR 716618 (84k:03125)
  • [9] J. Oxtoby, Measure and category, Springer-Verlag, Berlin, 1970. MR 584443 (81j:28003)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1264828-9
Keywords: Subsets of the plane, nonnull rectangles, closed null sets
Article copyright: © Copyright 1995 American Mathematical Society

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