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Ideals without ccc

Published online by Cambridge University Press:  12 March 2014

Marek Balcerzak
Affiliation:
Institute of Mathematics, Lódź Technical University, 90-924 Lodz, Poland, E-mail: mbalce@krysia.uni.lodz.pl
Andrzej RosŁanowski
Affiliation:
Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel Mathematical Institute of Wroclaw University, 50384 Wroclaw, Poland, E-mail: roslanow@math.huji.ac.il
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel 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).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1998

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