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Descriptive set theory of families of small sets
 J. Symbolic Logic
"... Abstract. This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σideals of closed or compact sets. ..."
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Abstract. This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σideals of closed or compact sets.
Trichotomies for ideals of compact sets
 J. SYMBOLIC LOGIC
"... We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal. ..."
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We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal.
NORMS ON POSSIBILITIES II: MORE CCC IDEALS ON 2 ω
"... Abstract. We use the method of norms on possibilities to answer a question of Kunen and construct a ccc σ–ideal on 2 ω with various closure properties and distinct from the ideal of null sets, the ideal of meager sets and their intersection. 628 revision:19970326 modified:19970326 0. Introductio ..."
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Abstract. We use the method of norms on possibilities to answer a question of Kunen and construct a ccc σ–ideal on 2 ω with various closure properties and distinct from the ideal of null sets, the ideal of meager sets and their intersection. 628 revision:19970326 modified:19970326 0. Introduction. In the present paper we use the method of norms on possibilities to answer a question of Kunen (see 0.1 below) and construct a ccc σ–ideal on 2 ω with various closure properties and distinct from the ideal of null sets, the ideal of meager sets and their intersection. The method we use is, in a sense, a generalization of the one studied systematically in
TOPOLOGICAL REPRESENTATIONS
"... Abstract. This paper studies the combinatorics of ideals which recently appeared in ergodicity results for analytic equivalence relations. The ideals have the following topological representation. There is a separable metrizable space X, a σideal I on X and a dense countable subset D of X such that ..."
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Abstract. This paper studies the combinatorics of ideals which recently appeared in ergodicity results for analytic equivalence relations. The ideals have the following topological representation. There is a separable metrizable space X, a σideal I on X and a dense countable subset D of X such that the ideal consists of those subsets of D whose closure belongs to I. It turns out that this definition is indepedent of the choice of D. We show that an ideal is of this form if and only if it is dense and countably separated. The latter is a variation of a notion introduced by Todorčević for gaps. As a corollary, we get that this class is invariant under the Rudin–Blass equivalence. This also implies that the space X can be always chosen to be compact so that I is a σideal of compact sets. We compute the possible descriptive complexities of such ideals and conclude that all analytic equivalence relations induced by such ideals are Π 0 3. We also prove that a coanalytic ideal is an intersection of ideals of this form if and only if it is weakly selective. 1.