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On some $\sigma $-ideal without ccc

Volume 158 / 2019

Marta Frankowska, Szymon Głąb Colloquium Mathematicum 158 (2019), 127-139 MSC: Primary 03E15; Secondary 03E20, 28E15. DOI: 10.4064/cm6991-8-2018 Published online: 29 July 2019


We prove that the $\sigma $-ideal $\sigma (a)$ generated by sets satisfying condition $(a)$ of M. Grande has property $(M)$, that is, there is a Borel function $f:\mathbb {R}\to 2^\mathbb {N}$ with $f^{-1}[\{x\}]\notin \sigma (a)$ for each $x\in 2^\mathbb {N}$, and consequently fails the ccc property. It is also shown that $\sigma (a)$ is generated by the family $\{E \setminus \varPhi (E) \colon E=\operatorname {cl}(E)\}$ where $\varPhi (E)$ is the set of density points of $E$. Finally, we show that for any $A \in \sigma (a)$ and $U$ open in the density topology, $A \cap U$ is meager in $U$.


  • Marta FrankowskaDepartment of Mathematics
    Gdańsk University
    Wita Stwosza 57
    80-952 Gdańsk, Poland
  • Szymon GłąbInstitute of Mathematics
    Łódź University of Technology
    Wólczańska 215
    93-005 Łódź, Poland

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