On some -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.
