Covering Property Axiom CPAcube and its consequences, Fund

In the paper we formulate a Covering Property Axiom CPAcube, 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 ∈ RR〉n<ω of Borel func-tions there are the sequences: 〈Pξ ⊂ R: ξ < ω1 〉 of compact sets and 〈Wξ ∈ [ω]ω: ξ < ω1 〉 such that R = ξ<ω1 Pξ and for every ξ < ω1: 〈fn Pξ〉n∈Wξ is a monotone uniformly convergent se-quence of uniformly continuous functions. (e) Total failure of Martin’s Axiom: c> ω1 and for every non-trivial ccc forcing P there exists ω1-many dense sets in P such that no filter intersects all of them. ∗AMS classification numbers: Primary 03E35; Secondary 03E17, 26A03. Key words and phrases: continuous images, perfectly meager, universally null, cofinalit