Uncountable γ - sets under axiom CPA gamecube

In the paper we formulate a Covering Property Axiom CPA cube , which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in R (which are strongly meager) as well as uncountable γ-sets in R which are not strongly meager. These sets must be of cardinality ω1 < c, since every γ-set is universally null, while CPA cube implies that every universally null has cardinality less than c = ω2. We will also show that CPA cube implies the existence of a partition of R into ω1 null compact sets. 1 Axiom CPA cube and other preliminaries Our set theoretic terminology is standard and follows that of [3]. 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 term Polish space for a complete separable metric space without isolated points. For a Polish space X, the symbol ∗AMS classification numbers: Primary 03E35; Secondary 03E17, 26A03.