On a rectangle inclusion problem

Pawlikowski, Janusz

doi:10.1090/S0002-9939-1995-1264828-9

On a rectangle inclusion problem
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Proc. Amer. Math. Soc. 123 (1995), 3189-3195 Request permission

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.

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