UNCOUNTABLE INTERSECTIONS OF OPEN SETS UNDER CPAprism

We prove that the Covering Property Axiom CPAprism, which holds in the iterated perfect set model, implies the following facts. • If G is an intersection of ω1-many open sets of a Polish space and G has cardinality continuum, then G contains a perfect set. • There exists a subset G of the Cantor set which is an intersection of ω1-many open sets but is not a union of ω1-many closed sets. The example from the second fact refutes a conjecture of Brendle, Larson, and Todorcevic. 1. Preliminaries and axiom CPAprism 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 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 C onto C . Axiom CPAprism was introduced by the authors in [5], where it is shown that it holds in the iterated perfect set model. Also, CPAprism is a simpler version of the axiom CPA, which is described in monograph [6]. (See also [4].) For the reader’s convenience, we will restate the axiom in the next few paragraphs. The main notions needed for the axiom are that of prism and prism-density. Let 0 < α < ω1 and let Φprism(α) be the family of all continuous injections f : C → C with the property that f(x) β = f(y) β ⇔ x β = y β for all β ∈ α and x, y ∈ C or, equivalently, such that for every β < α f β def = {〈x β, y β〉 : 〈x, y〉 ∈ f} is a one-to-one function from C into C . Functions f from Φprism(α) were first introduced, in a more general setting, in [8], where they are called projection-keeping 1991 Mathematics Subject Classification. Primary 03E35; Secondary 03E17.