Applications of the covering property axiom

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\u27 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 CPAgamecube of CPA there are uncountable strong gamma-sets on R . It is known that Martin\u27s Axiom (MA) implies the existence of a strong gamma-set on R . Our result is interesting since that CPAgamecube implies the negation of MA.;Next, we use the version CPAgameprism of CPA to construct some special ultrafilters on Q . An ultrafilter on Q is crowded provided it contains a filter basis consisting of perfect sets in Q . These ultrafilters have been constructed under various hypotheses. We study the properties of being P-point, Q-point, and o1-OK point and their negations; and prove under CPAgameprism the existence of an o1-generated crowded ultrafilter satisfying each consistent combination of these properties. We also refute an earlier claim by Ciesielski and Pawlikowski by proving under CPAgameprism that there are 2c -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 Ca . If X is a Polish space, we say that F ⊆ Perf( X ) is alpha-cube (alpha-prism) dense provided for every continuous injection f : Ca → X there exists a perfect cube (iterated perfect set) C ⊆ Ca such that f[C] ∈ F .;We prove that for every alpha \u3c o1 and every Polish space X there exists a family F such that F is beta-prism dense for every beta \u3c alpha but &vbm0;X\⋃ F&vbm0;=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