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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.
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.
AVOIDING FAMILIES AND TUKEY FUNCTIONS ON THE NOWHERE DENSE IDEAL
"... We investigate Tukey functions from the ideal of all closed nowhere dense subsets of 2 N. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of N. We also prove nonexistence of various special types ..."
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We investigate Tukey functions from the ideal of all closed nowhere dense subsets of 2 N. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of N. We also prove nonexistence of various special types of Tukey reductions from the nowhere dense ideal to analytic Pideals. In connection with these results, we study families F of clopen subsets of 2 N with the property that for each nowhere dense subset of 2 N there is a set in F not intersecting it. We call such families avoiding.
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.
Reviewed by Małgorzata Filipczak References
"... A note on Gδ ideals of compact sets. (English summary) ..."