Ideals without CCC

Marek Balcerzak, Andrzej RosŁanowski, Saharon Shelah

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

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 ⊆ P(X) of size c, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f : X → X 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 B ∉ I and a perfect set P ⊆ X for which the family {B + x : x ∈ 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 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 versions of (M) for invariant ideals on Polish groups are investigated (Section 6).

Original languageEnglish
Pages (from-to)128-147
Number of pages20
JournalJournal of Symbolic Logic
Volume63
Issue number1
DOIs
StatePublished - Mar 1998

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