We formulate a Covering Property Axiom , which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in ℝ (which are strongly meager) as well as uncountable γ-sets in ℝ which are not strongly meager. These sets must be of cardinality ω₁ < , since every γ-set is universally null, while implies that every universally null has cardinality less than = ω₂. We also show that implies the existence of a partition of ℝ into ω₁ null compact sets.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-2-3,
author = {Krzysztof Ciesielski and Andr\'es Mill\'an and Janusz Pawlikowski},
title = {Uncountable $\gamma$-sets under axiom $CPA\_{cube}^{game}$
},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {143-155},
zbl = {1020.03043},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-2-3}
}
Krzysztof Ciesielski; Andrés Millán; Janusz Pawlikowski. Uncountable γ-sets under axiom $CPA_{cube}^{game}$
. Fundamenta Mathematicae, Tome 177 (2003) pp. 143-155. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-2-3/