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) => 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).