Uncountable intersections of open sets under CPAprism

by

Krzysztof Ciesielski, and Janusz Pawlikowski

Proc. Amer. Math. Soc. 132(11) (2004), 3379-3385.

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 \omega1-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 \omega1-many open sets but is not a union of \omega1-many closed sets.
The example from the second fact refutes a conjecture of Brendle, Larson, and Todorcevic.\omega1<\continuum.