Covering Property Axiom $CPA_{cube}$ and its consequences

• Autorzy: Krzysztof Ciesielski , Janusz Pawlikowski
• Języki publikacji: EN

Abstrakty

EN

We formulate a Covering Property Axiom $CPA_{cube}$, which holds in the iterated perfect set model, and show that it implies easily the following facts.
(a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1].
(b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than 𝔠.
(c) cof(𝓝) = ω₁ < 𝔠, i.e., the cofinality of the measure ideal 𝓝 is ω₁.
(d) For every uniformly bounded sequence $⟨fₙ ∈ ℝ^{ℝ}⟩_{n<ω}$ of Borel functions there are sequences: $⟨P_ξ ⊂ ℝ: ξ < ω₁⟩$ of compact sets and $⟨W_ξ ∈ [ω]^ω: ξ < ω₁⟩$ such that $ℝ = ⋃_{ξ<ω₁}P_ξ$ and for every ξ < ω₁, $⟨fₙ ↾ P_ξ⟩_{n∈W_ξ}$ is a monotone uniformly convergent sequence of uniformly continuous functions.
(e) Total failure of Martin's Axiom: 𝔠 > ω₁ and for every non-trivial ccc forcing ℙ there exist ω₁ dense sets in ℙ such that no filter intersects all of them

Kategorie tematyczne

• 03E17: Cardinal characteristics of the continuum
• 26A03: Foundations: limits and generalizations, elementary topology of the line
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2003
• Tom: 176
• Numer: 1
• Strony: 63-75

Daty

wydano

2003

Twórcy

autor

Krzysztof Ciesielski

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

autor

Janusz Pawlikowski

• Department of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Identyfikatory

DOI

10.4064/fm176-1-5