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

Volume 176 / 2003

Krzysztof Ciesielski, Andrés Millán, Janusz Pawlikowski Fundamenta Mathematicae 176 (2003), 143-155 MSC: Primary 03E35; Secondary 03E17, 26A03. DOI: 10.4064/fm176-2-3

Abstract

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.

Authors

  • Krzysztof CiesielskiDepartment of Mathematics
    West Virginia University
    Morgantown, WV 26506-6310, U.S.A.
    e-mail
  • Andrés MillánDepartment of Mathematics
    West Virginia University
    Morgantown, WV 26506-6310, U.S.A.
    e-mail
  • Janusz PawlikowskiDepartment of Mathematics
    University of Wrocław
    Pl. Grunwaldzki 2/4
    50-384 Wrocław, Poland
    e-mail

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