$F_σ$-ideals and $ω_1 ω^*_1$-gaps in the Boolean algebras Ρ(ω)/I.

Mazur, Krzysztof

doi:10.4064/fm-138-2-103-111

Cited by 85 publications

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“…The motivation for this paper comes from the following open problem ([15] It is even unknown whether every F σδ ideal I has an F σ approximation K such that K∪K is meager (see [33]). A negative answer would imply that I is not countably determined by closed approximations.…”

Section: Problemsmentioning

confidence: 99%

Luzin gaps

Farah

2004

Trans. Amer. Math. Soc.

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Abstract. We isolate a class of F σδ ideals on N that includes all analytic P-ideals and all Fσ ideals, and introduce 'Luzin gaps' in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between quotient algebras over these ideals have continuous liftings. This gives a partial confirmation to the author's rigidity conjecture for quotients P(N)/I. We also prove that the ideals NWD(Q) and NULL(Q) have the Radon-Nikodým property, and (using OCA∞) a uniformization result for K-coherent families of continuous partial functions.One of the most fascinating facts about the Boolean algebra P(N)/ Fin was discovered by Hausdorff in 1908. In [20], he constructed two families A and B of sets of integers such that (a) A ∩ B is finite for all A ∈ A and all B ∈ B, (b) for every C ⊆ N either A \ C is infinite for some A ∈ A or B ∩ C is infinite for some B ∈ B, and (c) both and A and B have order-type equal to ω 1 with respect to the inclusion modulo finite. Families A and B satisfying (a) are orthogonal, those satisfying both (a) and (b) form a gap in P(N)/ Fin (or, they are not separated over Fin, the ideal of finite sets), and if they furthermore satisfy (c) they form an (ω 1 , ω 1 )-gap. If (a) and (b) hold and both A and B are linearly ordered by the inclusion modulo finite, a gap is linear. A gap (linear or not) is Hausdorff if both of its sides are σ-directed under the inclusion modulo finite.Another major advance in the study of gaps in P(N)/ Fin was made by Kunen ([29]), who used a condition originally isolated by Luzin in [32] to prove that for every (ω 1 , ω 1 )-gap there is a ccc poset that freezes it. (A gap is indestructible, or frozen, if it remains a gap in every ℵ 1 -preserving forcing extension.) Remarkably, a linear gap can be frozen by an ℵ 1 -preserving forcing if and only if it contains an (ω 1 , ω 1 )-gap. Kunen used freezing to prove that P(N)/ Fin need not be universal for linear orderings of size at most 2 ℵ0 even if Martin's Axiom, MA, is assumed. The technique of freezing gaps has played an important role in a variety of subjects, from automatic continuity in Banach algebras ([6]) to the study of automorphisms of P(N)/ Fin ([35]).

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Section: Problemsmentioning

confidence: 99%

Luzin gaps

Farah

2004

Trans. Amer. Math. Soc.

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“…An It is not hard to see that Fin(μ) is an F σ ideal. The following theorem of Mazur (see [14]) shows that each F σ ideal is of this form.…”

Section: Idealsmentioning

confidence: 96%

Adding ultrafilters by definable quotients

Hrušák¹,

Verner

2011

Rend. Circ. Mat. Palermo

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Forcing notions of the type P(ω)/I which do not add reals naturally add ultrafilters on ω. We investigate what classes of ultrafilters can be added in this way when I is a definable ideal. In particular, we show that if I is an F σ P-ideal the generic ultrafilter will be a P-point without rapid RK-predecessors which is not a strong P-point. This provides an answer to long standing open questions of Canjar and Laflamme.

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“…Namely, the incompatible equivalence relations of [2] are associated with S 0 2 P-ideals obtained from variations on Tsirelson's Banach space. The present paper, on the other hand, deals with c 0equalities (see De®nition 3.1), turbulent orbit equivalence relations that are P 0 3complete and include the family of equivalence relations constructed by Louveau and Velickovic in [20].…”

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confidence: 99%

Basis Problem for Turbulent Actions II: c ₀ -Equalities

Farah

2001

Proceedings of the London Mathematical Society

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This note is a contribution to the study of reducibility between Borel equivalence relations (see [14,11,9], or § 1). If E and F are Borel equivalence relations on Polish spaces X and Y, then we say that E is Borel reducible to F (in symbols,for all x; y P X. The early success in the study of the Borelreducibility was marked by the discovery of two dichotomy theorems by Silver [25] and Harrington, Kechris and Louveau [6]. For example, the latter result (socalled Glimm±Effros dichotomy, since it extends earlier theorems by J. Glimm and E. Effros) says that either a Borel equivalence relation E is smooth, that is, Borelreducible to the equality relation on some separable, completely metrizable (or Polish) space, or the Vitali equivalence relation, E 0 (see Example 3.2), is continuously reducible to E. Since E 0 is not smooth, this result can be interpreted as saying that E 0 is the minimal non-smooth Borel equivalence relation.Therefore the Glimm±Effros dichotomy gives one answer to the questioǹ When is an equivalence relation simply classi®able?' Another criterion for the simplicity of a Borel equivalence relation is that of being effectively reducible to the isomorphism relation of countable models. An equivalence relation E on a Polish space X is effectively reducible to the isomorphism relation of countable models if there is a Borel-measurable mapping f : X 3 2 N < N such that f x and f y code isomorphic countable structures if and only if x E y. This is a natural extension of smoothness which occurs in classical classi®cation results, like Cantor±Bendixson analysis, Ulm invariants, spectral theorems, . . . (see [9, § 0]).The ®rst example of a Borel equivalence relation which is not effectively reducible to the isomorphism relation of countable models was given by H. Friedman [5]. Later, Hjorth isolated the dynamical property of Polish group actions called turbulence (see [9,14]), and proved that in the realm of orbit equivalence relations arising from continuous Polish group actions,`effectively reducible to the isomorphism relation of countable models' is essentially equivalent with`induced by a turbulent action' [9, Theorem 0.14]. It is natural to ask whether a dichotomy along the lines of the Glimm±Effros dichotomy can be proved in the context of turbulence, namely whether there is the minimum turbulent orbit equivalence relation. The answer is known to be negative, since there are turbulent orbit equivalence relations E and F such that no turbulent orbit equivalence relation is Borel-reducible to both E and F (see [8]). We say that such E and F are incompatible. This still leaves a possibility that there is a ®nite list 2000 Mathematics Subject Classi®cation: 03E15.

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$F_σ$-ideals and $ω_1 ω^*_1$-gaps in the Boolean algebras Ρ(ω)/I.

Cited by 85 publications

References 0 publications

Luzin gaps

Luzin gaps

Adding ultrafilters by definable quotients

Basis Problem for Turbulent Actions II: c ₀ -Equalities

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$F_σ$-ideals and $ω_1 ω^*_1$-gaps in the Boolean algebras Ρ(ω)/I.

Cited by 85 publications

References 0 publications

Luzin gaps

Luzin gaps

Adding ultrafilters by definable quotients

Basis Problem for Turbulent Actions II: c 0 -Equalities

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Basis Problem for Turbulent Actions II: c ₀ -Equalities