On additive almost continuous functions under CPA game prism

Abstract. We prove that the Covering Property Axiom CPAgameprism, which holds in the iterated perfect set model, implies that there exists an additive discontinuous almost continuous function f: R → R whose graph is of measure zero. We also show that, under CPAgameprism, there exists a Hamel basis H for which, E+(H), the set of all linear combinations of elements from H with positive rational coefficients, is of measure zero. The existence of both of these examples follows from Martin’s axiom, while it is unknown whether either of them can be constructed in ZFC. As a tool for the constructions we will show that CPAgameprism implies its seemingly stronger version, in which ω1-many games are played simultaneously. 1. Preliminaries and axiom CPAgameprism 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 2ω will be denoted by a symbol C. We use term Polish space for a complete separable metric space without isolated points. For a Polish space X symbol Perf(X) will stand for a collection of all subsets of X homeomorphic to the Cantor set C. For a fixed 0 < α < ω1 and 0 < β ≤ α a symbol πβ will stand for the projection from Cα onto Cβ. In what follows we will consider R as a linear space over Q. For Z ⊂ R its linear span with respect to this structure will be denoted by LIN(Z). A subset H of R is a Hamel basis provided it is a linear basis of R over Q, that is, it is linearly independent and LIN(H) = R. Axiom CPAgameprism was introduced by the authors in [5], where it is shown that it holds in the iterated perfect set model. Also, CPAgameprism is a simpler version of the axiom CPA which is described in a monograph [9]. For the reader’s convenience, we will restate CPAgameprism in the next few paragraphs. For 0 < α < ω1 let Φprism(α) be the family of all continuous injections f: Cα → Cα with the property that f(x) β = f(y) β ⇔ x β = y β for all β ∈ α and x, y ∈ Cα. Functions Φprism(α) are called projection-keeping homeomorphisms. (Compare [11].