caa a22 4500 388039620 CHVBK 20180307125017.0 cr unu---uuuuu 161130e199803 xx s 000 0 eng 10.2307/2586592 doi S0022481200015371 pii (NATIONALLICENCE)cambridge-10.2307/2586592 Ideals without ccc [Elektronische Daten] 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 ϲ, 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 version's of (M) for invariant ideals on Polish groups are investigated (Section 6). Copyright © Association for Symbolic Logic 1998 Balcerzak Marek Institute of Mathematics, Lódź Technical University, 90-924 Lodz, Poland, E-mail: mbalce@krysia.uni.lodz.pl RosŁanowski Andrzej Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel Shelah Saharon Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel The Journal of Symbolic Logic Cambridge University Press 63/1(1998-03), 128-148 0022-4812 63:1<128 1998 63 JSL https://doi.org/10.2307/2586592 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586592 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Balcerzak Marek Institute of Mathematics, Lódź Technical University, 90-924 Lodz, Poland, E-mail: mbalce@krysia.uni.lodz.pl NATIONALLICENCE 700 1- RosŁanowski Andrzej Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel NATIONALLICENCE 700 1- Shelah Saharon Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/1(1998-03), 128-148 0022-4812 63:1<128 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge