A big symmetric planar set with small category projections


Krzysztof Ciesielski and Tomasz Natkaniec

Fund. Math. 178(3) (2003), 237-253.

We show that under appropriate set theoretic assumptions (which follow from Martin's axiom and the continuum hypothesis) there exists a nowhere meager subset A of R such that

  1. for each continuous nowhere constant function f: R-->R the set
    {c in R: proj[(f+c)\cap (AxA)] is not meager}
    is meager, and
  2. for each continuous f: R-->R the set
    {c in R: (f+c)\cap (AxA) is empty}
    is nowhere meager.
The existence of such a set follows also from the principle CPA, which holds in the iterated perfect set model. We also prove that the existence of a set A as in (1) cannot be proved in ZFC alone even when we restrict our attention to homeomorphisms of R. On the other hand, for the class of real analytic functions a Bernstein set A satisfying (2) exists in ZFC.

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Last modified October 24, 2003.