Applications of the Covering Property Axiom

Applications of the Covering Property Axiom Andrés Millán Millán The purpose of this work is two-fold. First, we present some consequences of the Covering Property Axiom CPA of Ciesielski and Pawlikowski which captures the combinatorial core of the Sacks’ model of the set theory. Second, we discuss the assumptions in the formulation of different versions of CPA. As our first application of CPA we prove that under the version CPA cube of CPA there are uncountable strong γ-sets on R. It is known that Martin’s Axiom (MA) implies the existence of a strong γ-set on R. Our result is interesting since that CPA cube implies the negation of MA. Next, we use the version CPA prism of CPA to construct some special ultrafilters on Q. An ultrafilter on Q is crowded provided it contains a filter basis consiting of perfect sets in Q. These ultrafilters have been constructed under various hypotheses. We study the properties of being P -point, Qpoint, and ω1-OK point and their negations, and prove under CPA game prism the existence of an ω1-generated crowded ultrafilter satisfying each consistent combination of these properties. We also refute an earlier claim by Ciesielski and Pawlikowski by proving under CPA prism that there are 2 -many crowded c-generated Q-points. We also study various notions of density, central to the foundation of CPA and defined in the set of all perfect subsets of a Polish space X . These notions involve the concepts of perfect cube and iterated perfect set on C. If X is a Polish space, we say that F ⊆ Perf(X ) is α-cube (α-prism) dense provided for every continuous injection f : C → X there exists a perfect cube (iterated perfect set) C ⊆ C such that f [C] ∈ F . We prove that for every α < ω1 and every Polish space X there exists a family F such that F is β-prism dense for every β < α but |X \⋃F| = c. Therefore, any attempt of strengthening of axiom CPAprism by replacing prism-density with any proper subclass of these densities leads to a false statement. The proof of this theorem is based in the following result: Any separately nowhere-constant function defined on a product of Polish spaces is one-to-one on some perfect cube.