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Ideals without ccc

  • Marek Balcerzak (a1), Andrzej RosŁanowski (a2) (a3) and Saharon Shelah (a2) (a4)

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).

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[1] M. Balcerzak , Can ideals without ccc be interesting?, Topology and its Applications, vol. 55 (1994), pp. 251260.

[3] M. Balcerzak and A. Rosłanowski , On Mycielski ideals, Proceedings of the American Mathematical Society, vol. 110 (1990), pp. 243250.

[5] K. J. Falconer , The geometry of fractal sets, Cambridge University Press, Cambridge, 1985.

[10] R. D. Mauldin , The Baire order of the functions continuous almost everywhere, Proceedings of the American Mathematical Society, vol. 41 (1973), pp. 535540.

[11] R. D. Mauldin , On the Borel subspaces of algebraic structures, Indiana University Mathematics Journal, vol. 29 (1980), pp. 261265.

[15] S. Shelah , On co-κ-Souslin relations, Israel Journal of Mathematics, vol. 47 (1984), pp. 139153.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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