A big symmetric planar set with small category projections

Krzysztof Ciesielski ¹   ; Tomasz Natkaniec ²

¹ Department of Mathematics West Virginia University Morgantown, WV 26506-6310, U.S.A.
² Department of Mathematics Gdańsk University Wita Stwosza 57 80-952 Gdańsk, Poland

Fundamenta Mathematicae, Tome 178 (2003) no. 3, pp. 237-253

Résumé

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.

Détail

DOI : 10.4064/fm178-3-4

Keywords: under appropriate set theoretic assumptions which follow martins axiom continuum hypothesis there exists nowhere meager set subset mathbb set mathbb colon cap times hbox meager meager each continuous nowhere constant function colon mathbb mathbb set mathbb colon cap times emptyset nowhere meager each continuous function colon mathbb mathbb existence set follows principle cpa which holds iterated perfect set model prove existence set cannot proved zfc alone even restrict attention homeomorphisms mathbb other class real analytic functions bernstein set satisfying exists zfc