Uncountable $\gamma $-sets under axiom ${\rm CPA}_{\rm cube}^{\rm game}$

Krzysztof Ciesielski ¹   ; Andrés Millán ¹   ; Janusz Pawlikowski ²

¹ Department of Mathematics West Virginia University Morgantown, WV 26506-6310, U.S.A.
² Department of Mathematics University of Wrocław Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Fundamenta Mathematicae, Tome 176 (2003) no. 2, pp. 143-155

Résumé

We formulate a Covering Property Axiom CPA$_{\rm cube}^{\rm game}$, which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong $\gamma $-sets in ${\mathbb R}$ (which are strongly meager) as well as uncountable $\gamma $-sets in ${\mathbb R}$ which are not strongly meager. These sets must be of cardinality $\omega _1{\mathfrak c}$, since every $\gamma $-set is universally null, while CPA$_{\rm cube}^{\rm game}$ implies that every universally null has cardinality less than ${\mathfrak c}=\omega _2$. We also show that CPA$_{\rm cube}^{\rm game}$ implies the existence of a partition of ${\mathbb R}$ into $\omega _1$ null compact sets.

Détail

DOI : 10.4064/fm176-2-3

Keywords: formulate covering property axiom cpa cube game which holds iterated perfect set model implies existence uncountable strong gamma sets mathbb which strongly meager uncountable gamma sets mathbb which strongly meager these sets cardinality omega mathfrak since every gamma set universally null while cpa cube game implies every universally null has cardinality mathfrak omega cpa cube game implies existence partition mathbb omega null compact sets