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

Krzysztof Ciesielski ; Janusz Pawlikowski

Fundamenta Mathematicae, Tome 177 (2003), p. 63-75 / Harvested from The Polish Digital Mathematics Library

Résumé

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) cof() = ω₁ < , i.e., the cofinality of the measure ideal 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: > ω₁ and for every non-trivial ccc forcing ℙ there exist ω₁ dense sets in ℙ such that no filter intersects all of them

Détail

Publié le : 2003-01-01

Zbl 1013.03057

EUDML-ID : urn:eudml:doc:283393