Continuous images of big sets and additivity of s 0 under CPA prism

We prove that the Covering Property Axiom CPAprism, which holds in the iterated perfect set model, implies the following facts. • There exists a family G of uniformly continuous functions from R to [0, 1] such that |G| = ω1 and for every S ∈ [R] there exists a g ∈ G with g[S] = [0, 1]. • The additivity of the Marczewski’s ideal s0 is equal to ω1 < c. 1 Preliminaries and axiom CPAprism Our set theoretic terminology is standard and follows that of [1]. In particular, |X| stands for the cardinality of a set X and c = |R|. The Cantor set 2 will be denoted by a symbol C. We use the term Polish space for a complete separable metric space without isolated points. For a Polish space X the symbol Perf(X) will stand for the collection of all subsets of X homeomorphic to the Cantor set C. For a fixed 0 < α < ω1 and 0 < β ≤ α the symbol πβ will stand for the projection from C onto C . Axiom CPAprism was introduced by the authors in [3], where it is shown that it holds in the iterated perfect set model. Also, CPAprism is a simpler