We study the structure of analytic ideals of subsets of the natural numbers. For example, we prove that for an analytic ideal I, either the ideal {X (Ω × Ω: En X ({0, 1,…,n} × Ω } is Rudin-Keisler below I, or I is very simply induced by a lower semicontinuous submeasure. Also, we show that the class of ideals induced in this manner by lsc submeasures coincides with Polishable ideals as well as analytic P-ideals. We study this class of (...) ideals and characterize, for example, when the ideals in it are Fσ or when they carry a locally compact group topology. We apply these results to Borel partial orders to rederive a theorem of Todorcevic and to Borel equivalence relations to answer a question of Kechris and Louveau. As another application we give a characterization of σ-ideals of μ-zero sets for Maharam submeasures μ on the Cantor set which is to a large extent analogous to a characterization of the meager ideal due to Kechris and the author. (shrink)

§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters. There is also a substantial interest in nicely definable ideals—these by old results of Sierpiński are very far from being maximal— and the structure of such ideals will concern us in this announcement. In addition to being interesting in their own right, Borel and analytic ideals occur naturally in (...) the investigations of compact subsets of the space of all Baire class 1 functions on a Polish space, see [12, 18]. Also, certain objects associated with such ideals are of considerable interest and were quite extensively studied by several authors. Let us list here three examples; in all three of them I stands for an analytic or Borel ideal.1. The partial order induced by I on P: X ≥I Y iff X \ Y ϵ I and the partial order.2. Boolean algebras of the form P/I and their automorphisms.3. The equivalence relation associated with I: XEI Y iff X Δ ϵ I.In Section 4, we will have an opportunity to state some consequences of our results for equivalence relations as in 3. (shrink)

The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L of T, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well (...) as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic1 305–319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math.231 605–623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint, arXiv:1201.5221v1]. (shrink)

We investigate the Tukey order in the class of Fσ ideals of subsets of ω. We show that no nontrivial Fσ ideal is Tukey below a Gδ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we (...) show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets. (shrink)

We prove that for every family I of closed subsets of a Polish space each Σ 1 1 set can be covered by countably many members of I or else contains a nonempty Π 0 2 set which cannot be covered by countably many members of I. We prove an analogous result for κ-Souslin sets and show that if A ♯ exists for any $A \subset \omega^\omega$ , then the above result is true for Σ 1 2 sets. A theorem (...) of Martin is included stating that this result is also true for weakly homogeneously Souslin sets. As an application of our results we derive from them a general form of Hurewicz's theorem due to Kechris, Louveau, and Woodin and a theorem of Feng on the open covering axiom. Also some well-known theorems on finding "big" closed sets inside of "big" Σ 1 1 and Σ 1 2 sets are consequences of our results. (shrink)

We prove that every analytic set in ω ω × ω ω with σ-bounded sections has a not σ-bounded closed free set. We show that this result is sharp. There exists a closed set with bounded sections which has no dominating analytic free set, and there exists a closed set with non-dominating sections which does not have a not σ-bounded analytic free set. Under projective determinacy analytic can be replaced in the above results by projective.

We discuss in the paper the following problem: Given a function in a given Baire class, into "how many" (in terms of cardinal numbers) functions of lower classes can it be decomposed? The decomposition is understood here in the sense of the set-theoretical union.

We prove that there is a -ideal of compact sets which is strictly above in the Tukey order. Here is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Veličković asked in [Alain Louveau, Boban Veličković, Analytic ideals and cofinal types, Ann. Pure Appl. Logic 99 171–195].

We show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, (...) we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces. (shrink)

We prove several trichotomy results for ideals of compact sets. Typically, we show that a "sufficiently rich" universally Baire ideal is either $\Pi _{3}^{0}$-hard, or $\Sigma _{3}^{0}$-hard, or else a σ-ideal.

We give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$-free structures, and (...) the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics. (shrink)