Uncountable \gamma-sets under axiom CPAcubegame

by

Krzysztof Ciesielski, Andrés Millán Millán, and Janusz Pawlikowski

Fund. Math. 176(1) (2003), 143-155.

In the paper we formulate a Covering Property Axiom CPAcubegame, which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong \gamma-sets in R (which are strongly meager) as well as uncountable \gamma-sets in R are not strongly meager. These sets must be of cardinality \omega1, since every \gamma-set is universally null, while CPAcubegame implies that every universally null has cardinality less than \continuum=\omega2.

We will also show that CPAcubegame implies the existence of a partition of R into \omega1 null compact sets.