On some $\sigma $-ideal without ccc
Volume 158 / 2019
Colloquium Mathematicum 158 (2019), 127-139
MSC: Primary 03E15; Secondary 03E20, 28E15.
DOI: 10.4064/cm6991-8-2018
Published online: 29 July 2019
Abstract
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 $U$.
