Uncountable intersections of open sets under CPA_prism
by
Krzysztof Ciesielski
West Virginia University
Coauthors: Janusz Pawlikowski

We prove that the Covering Property Axiom CPA_prism, which holds in the iterated perfect set model, implies the following facts.

(1) If G is an intersection of omega_1-many open sets of a Polish space and G has cardinality continuum then G contains a perfect set.

(2) There exists a subset G of the Cantor set which is an intersection of omega_1-many open sets but is not a union of omega_1-many closed sets.

(3) There exists a family G of uniformly continuous functions from R to [0, 1] such that |G|=omega_1 and for every subset S of R of cardinality continuum there exists a g in G with g[S]=[0, 1].

The example (2) refutes a conjecture of Brendle, Larson, and Todorcevic. The arguments for (2) and (3) are closely related.

Date received: March 5, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakf-05.