A big symmetric planar set with small category projections

• Autorzy: Krzysztof Ciesielski , Tomasz Natkaniec
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 03E50: Continuum hypothesis and Martin's axiom
• 26A99: None of the above, but in this section
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2003
• Tom: 178
• Numer: 3
• Strony: 237-253

Daty

wydano

2003

Twórcy

autor

Krzysztof Ciesielski

• Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, U.S.A.

autor

Tomasz Natkaniec

• Department of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952 Gdańsk, Poland

Identyfikatory

DOI

10.4064/fm178-3-4