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Ideals without ccc

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

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).

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[1]Balcerzak, M., Can ideals without ccc be interesting?, Topology and its Applications, vol. 55 (1994), pp. 251260.
[2]Balcerzak, M., Functions with fibres large on each nonvoid open set, Acta Universitatis Lodziensis, Folia Mathematica, vol. 7 (1995), pp. 310.
[3]Balcerzak, M. and Rosłanowski, A., On Mycielski ideals, Proceedings of the American Mathematical Society, vol. 110 (1990), pp. 243250.
[4]Bruckner, A., Differentiation of real functions, Lecture notes in mathematics, vol. 659, Springer-Verlag, Berlin, 1978.
[5]Falconer, K. J., The geometry of fractal sets, Cambridge University Press, Cambridge, 1985.
[6]Harrington, L. and Shelah, S., Counting equivalence classes for co-κ-Souslin equivalence relations, Logic colloquium 1980 (van Dalen, D., Lascar, D., and Smiley, J., editors), North Holland, 1982.
[7]Jech, T., Set theory, Academic Press, New York, 1978.
[8]Judah, H. and Rosianowski, A., Martin's axiom and the continuum, this Journal, vol. 60 (1995), pp. 374392.
[9]Judah, H. and Shelah, S., sets of reals, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 207223.
[10]Mauldin, R. D., The Baire order of the functions continuous almost everywhere, Proceedings of the American Mathematical Society, vol. 41 (1973), pp. 535540.
[11]Mauldin, R. D., On the Borel subspaces of algebraic structures, Indiana University Mathematics Journal, vol. 29 (1980), pp. 261265.
[12]Mycielski, J., Independent sets in topological algebras, Fundementa Mathematicae, vol. 55 (1964), pp. 139147.
[13]Mycielski, J., Algebraic independence and measure, Fundamenta Mathematicae, vol. 61 (1967), pp. 165169.
[14]Mycielski, J., Some new ideals of sets on the real line, Colloquium Mathematicum, vol. 20 (1969), pp. 7176.
[15]Shelah, S., On co-κ-Souslin relations, Israel Journal of Mathematics, vol. 47 (1984), pp. 139153.

Ideals without ccc

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

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