Ideals without ccc
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 c, consisting of Borel sets which are not in I. Condition (M) states that there is a function f:X> 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) => (M) => (B) => notccc can hold. We build a sigmaideal on the Cantor group witnessing''(M) and not (D)'' (Section 2). A modified version of that sigmaideal 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).
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 1996
 DOI:
 10.48550/arXiv.math/9610219
 arXiv:
 arXiv:math/9610219
 Bibcode:
 1996math.....10219B
 Keywords:

 Mathematics  Logic
 EPrint:
 J. Symbolic Logic 63 (1998), 128147