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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.
A Gδ Ideal of Compact Sets Strictly Above the Nowhere Dense
 Ideal in the Tukey Order, Ann. Pure Appl. Logic 156 (2008
"... Abstract. We prove that there is a Gδ σideal of compact sets which is strictly above NWD in the Tukey order. Here NWD is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Veličkovic ́ asked in [4]. 1. ..."
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Abstract. We prove that there is a Gδ σideal of compact sets which is strictly above NWD in the Tukey order. Here NWD is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Veličkovic ́ asked in [4]. 1.
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.
KENILWORTH
, 1997
"... We construct a Gδ σideal I of compact subsets of 2ω such that I contains all the singletons but there is no dense Gδ set D ⊆ 2ω such that {K ⊆ D: K compact} ⊆ I. This answers a question of A. S. Kechris in the negative. ..."
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We construct a Gδ σideal I of compact subsets of 2ω such that I contains all the singletons but there is no dense Gδ set D ⊆ 2ω such that {K ⊆ D: K compact} ⊆ I. This answers a question of A. S. Kechris in the negative.