A combinatorial core of the iterated perfect set model
by
Krzysztof Chris Ciesielski
Department of Mathematics, West Virginia University
Coauthors: Janusz Pawlikowski (University of Wroclw and West Virginia University)

Many interesting mathematical properties, especially concerning real analysis, are known to be true in the iterated perfect set (Sacks) model, while they are false under the continuum hypothesis. However, the proofs that these facts are indeed true in this model are usually very technical and involve heavy forcing machinery. In the presented paper we extract a combinatorial principle, a Covering Property Axiom CPA, that is true in the model and show that it implies, in particular, the following facts that are false under CH. (Most to these were known earlier to be true on the iterated perfect set model. However our proofs are essentially simpler than the original arguments.)

• For every subset S of R of cardinality 2^\omega there exists a (uniformly) continuous function f:R --> [0, 1] such that f[S]=[0, 1].
• Every perfectly meager set S subset R has cardinality less than 2^\omega.
• Every universally null set S subset R has cardinality less than 2^\omega.
• The cofinality of the measure ideal is less than 2^\omega.
• Every selective ultrafilter on \omega is generated by less than 2^\omega many sets.
• There is no Darboux Sierpi\'nski-Zygmund function f:R --> R, that is, for every Darboux function f:R --> R there is a subset Y of R of cardinality 2^\omega such that f|Y is continuous.
• For every Darboux function g:R --> R there exists a continuous nowhere constant function f:R --> R such that f+g is Darboux.
• The plane R² can be covered by less than 2^\omega many sets each of which is a graph of a partial function defined and differentiable on a perfect subset of either horizontal or vertical axis.

Date received: February 28, 2000