# Sh:512

- Balcerzak, M., Rosłanowski, A., & Shelah, S. (1998).
*Ideals without ccc*. J. Symbolic Logic,**63**(1), 128–148. arXiv: math/9610219 DOI: 10.2307/2586592 MR: 1610790 -
Abstract:

Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F\subseteq P(X) of size {\bf c}, consisting of Borel sets which are not in I. Condition (M) states that there is a function f:X\rightarrow X with f^{-1}[\{x\}]\notin I for each x\in X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B\notin I and a perfect set P\subseteq X for which the family \{ B+x: x\in P\} is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D)\Rightarrow (M)\Rightarrow (B)\Rightarrow not-ccc can hold. We build a \sigma-ideal on the Cantor group witnessing "(M) and not (D)" (Section 2). A modified version of that \sigma-ideal contains the whole space (Section 3). Some consistency results deriving (M) from (B) for "nicely" defined ideals are established (Section 4). We show that both ccc and (M) can fail (Theorems 1.3 and 4.2). Finally, some sharp versions of (M) for invariant ideals on Polish groups are investigated (Section 5). - published version (22p)

Bib entry

@article{Sh:512, author = {Balcerzak, Marek and Ros{\l}anowski, Andrzej and Shelah, Saharon}, title = {{Ideals without ccc}}, journal = {J. Symbolic Logic}, fjournal = {The Journal of Symbolic Logic}, volume = {63}, number = {1}, year = {1998}, pages = {128--148}, issn = {0022-4812}, doi = {10.2307/2586592}, mrclass = {03E15 (03E35 03E55 54H05)}, mrnumber = {1610790}, mrreviewer = {Miroslav Repick\'y}, doi = {10.2307/2586592}, note = {\href{https://arxiv.org/abs/math/9610219}{arXiv: math/9610219}}, arxiv_number = {math/9610219} }