A big symmetric planar set with small category projections

Tom 178 / 2003

Krzysztof Ciesielski, Tomasz Natkaniec Fundamenta Mathematicae 178 (2003), 237-253 MSC: Primary 03E35; Secondary 26A99, 03E50. DOI: 10.4064/fm178-3-4


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

(i) the set $\{c\in{\mathbb R}\colon\, \pi[({f+c}) \cap (A\times A)]\hbox{ is not meager}\}$ is meager for each continuous nowhere constant function $f\colon\,{\mathbb R}\to{\mathbb R}$,

(ii) the set $\{c\in{\mathbb R}\colon\, (f+c)\cap (A\times A)=\emptyset\}$ is nowhere meager for each continuous function $f\colon\,{\mathbb R}\to{\mathbb R}$.

The existence of such a set also follows from the principle CPA, which holds in the iterated perfect set model. We also prove that the existence of a set $A$ as in (i) cannot be proved in ZFC alone even when we restrict our attention to homeomorphisms of $\mathbb R$. On the other hand, for the class of real-analytic functions a Bernstein set $A$ satisfying (ii) exists in ZFC.


  • Krzysztof CiesielskiDepartment of Mathematics
    West Virginia University
    Morgantown, WV 26506-6310, U.S.A.
  • Tomasz NatkaniecDepartment of Mathematics
    Gdańsk University
    Wita Stwosza 57
    80-952 Gdańsk, Poland

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