The Journal of Symbolic Logic

Research Article

Ideals without ccc

Marek Balcerzaka1, Andrzej RosŁanowskia2a3 and Saharon Shelaha4a5

a1 Institute of Mathematics, Lódź Technical University, 90-924 Lodz, Poland, E-mail: mbalce@krysia.uni.lodz.pl

a2 Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

a3 Mathematical Institute of Wroclaw University, 50384 Wroclaw, Poland, E-mail: roslanow@math.huji.ac.il

a4 Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

a5 Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA, E-mail: shelah@math.huji.ac.il

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 FP(X) of size ϲ, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f : XX with f −1[{x}] ∉ I for each x ∈ X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set BI and a perfect set PX for which the family {B+x : xP} 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 σ-ideal on the Cantor group witnessing (M) & ¬(D) (Section 2). A modified version of that σ-ideal contains the whole space (Section 3). Some consistency results on deriving (M) from (B) for “nicely” defined ideals are established (Sections 4 and 5). We show that both ccc and (M) can fail (Theorems 1.3 and 5.6). Finally, some sharp version's of (M) for invariant ideals on Polish groups are investigated (Section 6).

(Received March 11 1994)

(Revised July 25 1996)