A big symmetric planar set with small category projections

Krzysztof Ciesielski, and Tomasz Natkaniec. "A big symmetric planar set with small category projections." Fundamenta Mathematicae 178.3 (2003): 237-253. <http://eudml.org/doc/283346>.

@article{KrzysztofCiesielski2003,
abstract = { 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 ⊂ ℝ such that (i) the set \{c ∈ ℝ: π[(f+c) ∩ (A×A)] is not meager\} is meager for each continuous nowhere constant function f: ℝ → ℝ, (ii) the set \{c ∈ ℝ: (f+c) ∩ (A×A) = ∅\} is nowhere meager for each continuous function f: ℝ → 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 ℝ. On the other hand, for the class of real-analytic functions a Bernstein set A satisfying (ii) exists in ZFC. },
author = {Krzysztof Ciesielski, Tomasz Natkaniec},
journal = {Fundamenta Mathematicae},
keywords = {-category projections; transfinite induction; nowhere meager sets; Covering Property Axiom CPA; -oracle},
language = {eng},
number = {3},
pages = {237-253},
title = {A big symmetric planar set with small category projections},
url = {http://eudml.org/doc/283346},
volume = {178},
year = {2003},
}

TY - JOUR
AU - Krzysztof Ciesielski
AU - Tomasz Natkaniec
TI - A big symmetric planar set with small category projections
JO - Fundamenta Mathematicae
PY - 2003
VL - 178
IS - 3
SP - 237
EP - 253
AB - 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 ⊂ ℝ such that (i) the set {c ∈ ℝ: π[(f+c) ∩ (A×A)] is not meager} is meager for each continuous nowhere constant function f: ℝ → ℝ, (ii) the set {c ∈ ℝ: (f+c) ∩ (A×A) = ∅} is nowhere meager for each continuous function f: ℝ → 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 ℝ. On the other hand, for the class of real-analytic functions a Bernstein set A satisfying (ii) exists in ZFC.
LA - eng
KW - -category projections; transfinite induction; nowhere meager sets; Covering Property Axiom CPA; -oracle
UR - http://eudml.org/doc/283346
ER -