Ideals without ccc

1998 ◽  
Vol 63 (1) ◽  
pp. 128-148 ◽  
Author(s):  
Marek Balcerzak ◽  
Andrzej RosŁanowski ◽  
Saharon Shelah
Keyword(s):  
Polish Space ◽  
Borel Function ◽  
Borel Set ◽  
Borel Sets ◽  
Polish Groups ◽  
Perfect Set ◽  
Consistency Results ◽  
The Family ◽  
Whole Space ◽  
Condition B

AbstractLet I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F ⊆ P(X) of size ϲ, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f : X → X with f−1[{x}] ∉ I for each x ∈ X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B ∉ I and a perfect set P ⊆ X for which the family {B+x : x ∈ P} is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) ⇒ (M) ⇒ (B) ⇒ not-ccc can hold. We build a σ-ideal on the Cantor group witnessing (M) & ¬(D) (Section 2). A modified version of that σ-ideal contains the whole space (Section 3). Some consistency results on deriving (M) from (B) for “nicely” defined ideals are established (Sections 4 and 5). We show that both ccc and (M) can fail (Theorems 1.3 and 5.6). Finally, some sharp version's of (M) for invariant ideals on Polish groups are investigated (Section 6).

scholarly journals Forcing Properties of Ideals of Closed Sets

2011 ◽  
Vol 76 (3) ◽  
pp. 1075-1095 ◽  
Author(s):  
Marcin Sabok ◽  
Jindřich Zapletal
Keyword(s):  
Polish Space ◽  
Borel Function ◽  
Constant Property ◽  
Borel Set ◽  
Borel Sets ◽  
Combinatorial Properties ◽  
Closed Sets ◽  
Cohen Real ◽  
The Ideal

AbstractWith every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ -ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal.We also study the 1–1 or constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1–1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P1 adds a Cohen real, or else I has the 1–1 or constant property.

scholarly journals Analytic equivalence relations and Ulm-type classifications

1995 ◽  
Vol 60 (4) ◽  
pp. 1273-1300 ◽  
Author(s):  
Greg Hjorth ◽  
Alexander S. Kechris
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Borel Measure ◽  
Polish Space ◽  
Borel Function ◽  
Equivalence Relations ◽  
Cantor Space ◽  
Continuous Action ◽  
Borel Set ◽  
Continuous Embedding

Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.The original Glimm-Effros dichotomy, established by Effros [Ef], [Ef1], who generalized work of Glimm [G1], asserts that if an Fσ equivalence relation on a Polish space X is induced by the continuous action of a Polish group G on X, then exactly one of the following alternatives holds:(I) Elements of X can be classified up to E-equivalence by “concrete invariants” computable in a reasonably definable way, i.e., there is a Borel function f: X → Y, Y a Polish space, such that xEy ⇔ f(x) = f(y), or else(II) E contains a copy of a canonical equivalence relation which fails to have such a classification, namely the relation xE0y ⇔ ∃n∀m ≥ n(x(n) = y(n)) on the Cantor space 2ω (ω = {0,1,2, …}), i.e., there is a continuous embedding g: 2ω → X such that xE0y ⇔ g(x)Eg(y).Moreover, alternative (II) is equivalent to:(II)′ There exists an E-ergodic, nonatomic probability Borel measure on X, where E-ergodic means that every E-invariant Borel set has measure 0 or 1 and E-nonatomic means that every E-equivalence class has measure 0.

scholarly journals THE FINE STRUCTURE OF THE INTUITIONISTIC BOREL HIERARCHY

2009 ◽  
Vol 2 (1) ◽  
pp. 30-101 ◽  
Author(s):  
WIM VELDMAN
Keyword(s):  
Fine Structure ◽  
Infinite Sequence ◽  
Polish Space ◽  
Main Tool ◽  
Inductive Definition ◽  
Borel Set ◽  
Borel Sets ◽  
Continuity Principle ◽  
Borel Hierarchy ◽  
Axiom Of Countable Choice

In intuitionistic analysis, a subset of a Polish space like ℝ or ${\cal N}$ is called positively Borel if and only if it is an open subset of the space or a closed subset of the space or the result of forming either the countable union or the countable intersection of an infinite sequence of (earlier constructed) positively Borel subsets of the space. The operation of taking the complement is absent from this inductive definition, and, in fact, the complement of a positively Borel set is not always positively Borel itself (see Veldman, 2008a). The main result of Veldman (2008a) is that, assuming Brouwer's Continuity Principle and an Axiom of Countable Choice, one may prove that the hierarchy formed by the positively Borel sets is genuinely growing: every level of the hierarchy contains sets that do not occur at any lower level. The purpose of the present paper is a different one: we want to explore the truly remarkable fine structure of the hierarchy. Brouwer's Continuity Principle again is our main tool. A second axiom proposed by Brouwer, his Thesis on Bars is also used, but only incidentally.

The topological Vaught's conjecture and minimal counterexamples

1994 ◽  
Vol 59 (3) ◽  
pp. 757-784 ◽  
Author(s):  
Howard Becker
Keyword(s):  
Topological Group ◽  
Polish Space ◽  
Sufficient Conditions ◽  
Usual Model ◽  
Borel Set ◽  
Isomorphism Classes ◽  
Perfect Set ◽  
Minimal Counterexample ◽  
Necessary And Sufficient ◽  
Borel Measurable

Let G be a Polish topological group, let X be a Polish space, let J: G × X → X be a Borel-measurable action of G on X, and let A ⊂ X be a Borel set which is invariant with respect to J, i.e., a Borel set of orbits. The following statement, or various equivalent versions of it, is known as the Topological Vaught's Conjecture.Let (G, X, J, A) be as above. Either A contains only countably many orbits, or else, A contains perfectly many orbits.We say that A contains perfectly many orbits if there is a perfect set P ⊂ A such that no two elements of P are in the same orbit. (Assuming ¬CH, A contains perfectly many orbits iff it contains 2ℵ0 orbits.) The Topological Vaught's Conjecture implies the usual, model theoretic, Vaught's Conjecture for Lω1ω, since the isomorphism classes are the orbits of an action of the group of permutations of ω; we give details in §0. The “Borel” assumption cannot be weakened for either A or J.Given a Borel-measurable Polish action (G,X,J) and an invariant Borel set B ⊂ X, we say that B is a minimal counterexample if (G,X,J,B) is a counterexample to the Topological Vaught's Conjecture and for every invariant Borel C ⊂ B, either C or B\C contains only countably many orbits. This paper is concerned with counterexamples to the Topological Vaught's Conjecture (of course, there may not be any), and in particular, with minimal counterexamples. First, there is a theorem on the existence of minimal counterexamples. This theorem was known for the model theoretic case (it is due to Harnik and Makkai), and is here generalized to arbitrary Borel-measurable Polish actions. Second, we study the properties of minimal counterexamples. We give two different necessary and sufficient conditions for a counterexample to be minimal, as well as some consequences of minimality. Some of these results are proved assuming determinacy axioms.This second part seems to be new even in the model theoretic case.

scholarly journals On $$\psi _{{\mathcal{H}}}( . )$$-operator in weak structure spaces with hereditary classes

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
H. M. Abu-Donia ◽  
Rodyna A. Hosny
Keyword(s):  
Topological Spaces ◽  
Structure Space ◽  
Hereditary Class ◽  
The Family ◽  
Main Target ◽  
Whole Space ◽  
Weak Structure

Abstract Weak structure space (briefly, wss) has master looks, when the whole space is not open, and these classes of subsets are not closed under arbitrary unions and finite intersections, which classify it from typical topology. Our main target of this article is to introduce $$\psi _{{\mathcal {H}}}(.)$$ ψ H ( . ) -operator in hereditary class weak structure space (briefly, $${\mathcal {H}}wss$$ H w s s ) $$(X, w, {\mathcal {H}})$$ ( X , w , H ) and examine a number of its characteristics. Additionally, we clarify some relations that are credible in topological spaces but cannot be realized in generalized ones. As a generalization of w-open sets and w-semiopen sets, certain new kind of sets in a weak structure space via $$\psi _{{\mathcal {H}}}(.)$$ ψ H ( . ) -operator called $$\psi _{{\mathcal {H}}}$$ ψ H -semiopen sets are introduced. We prove that the family of $$\psi _{{\mathcal {H}}}$$ ψ H -semiopen sets composes a supra-topology on X. In view of hereditary class $${\mathcal {H}}_{0}$$ H 0 , $$w T_{1}$$ w T 1 -axiom is formulated and also some of their features are investigated.

Constructible lattices of c-degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 739-754
Author(s):  
C.P. Farrington
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Combinatorial Complexity ◽  
Generic Extension ◽  
Transitive Model ◽  
Perfect Set ◽  
Consistency Results ◽  
Combinatorial Principles ◽  
Souslin Trees

This paper is devoted to the proof of the following theorem.Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that(i) N and M have the same cardinals, and κN ⊂ M;(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;(iii) if A ⊂ On and A Є N, there is B Є P(κ+)N such that L(A) = L(B).The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.

References and Remarks on Rotating Fluids

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Initial Data ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Incompressible Limit ◽  
Rotating Fluids ◽  
Inviscid Fluids ◽  
First Case ◽  
The Family ◽  
Whole Space

The problem investigated in this part can be seen as a particular case of the study of the asymptotic behavior (when ε tends to 0) of solutions of systems of the type where Δε is a non-negative operator of order 2 possibly depending on ε, and A is a skew-symmetric operator. This framework contains of course a lot of problems including hyperbolic cases when Δε = 0. Let us notice that, formally, any element of the weak closure of the family (uε)ε>0 belongs to the kernel of A. We can distinguish from the beginning two types of problems depending on the nature of the initial data. The first case, known as the well-prepared case, is the case when the initial data belong to the kernel of A. The second case, known as the ill-prepared case, is the general case. In the well-prepared case, let us mention the pioneer paper by S. Klainerman and A. Majda about the incompressible limit for inviscid fluids. A lot of work has been done in this case. In the more specific case of rotating fluids, let us mention the work by T. Beale and A. Bourgeois and T. Colin and P. Fabrie. In the case of ill-prepared data, the nature of the domain plays a crucial role. The first result in this case was established in 1994 in the pioneering work by S. Schochet for periodic boundary conditions. In the context of general hyperbolic problems, he introduced the key concept of limiting system (see the definition on page 125). In the more specific case of viscous rotating fluids, E. Grenier proved in 1997 in Theorem 6.3, page 125, of this book. At this point, it is impossible not to mention the role of the inspiration played by the papers by J.-L. Joly, G. Métivier and J. Rauch (see for instance and). In spite of the fact that the corresponding theorems have been proved afterwards, the case of the whole space, the purpose of Chapter 5 of this book, appears to be simpler because of the dispersion phenomena.

Compositions of Set Operations

1970 ◽  
Vol 22 (2) ◽  
pp. 227-234
Author(s):  
D. W. Bressler ◽  
A. H. Cayford
Keyword(s):  
Topological Structure ◽  
Borel Sets ◽  
Set Operations ◽  
The Family ◽  
Analytic Sets

The set operations under consideration are Borel operations and Souslin's operation (). With respect to a given family of sets and in a setting free of any topological structure there are defined three Borel families (Definitions 3.1) and the family of Souslin sets (Definition 4.1). Conditions on an initial family are determined under which iteration of the Borel operations with Souslin's operation () on the initial family and the families successively produced results in a non-decreasing sequence of families of analytic sets (Theorem 5.2.1 and Definition 3.5). A classification of families of analytic sets with respect to an initial family of sets is indicated in a manner analogous to the familiar classification of Borel sets (Definition 5.3).

Small Coverings with Smooth Functions under the Covering Property Axiom

2005 ◽  
Vol 57 (3) ◽  
pp. 471-493 ◽  
Author(s):  
Krzysztof Ciesielski ◽  
Janusz Pawlikowski
Keyword(s):  
Borel Function ◽  
Covering Property ◽  
Smooth Functions ◽  
Differentiable Functions ◽  
Perfect Set ◽  
Covering Property Axiom

AbstractIn the paper we formulate a Covering Property Axiom, CPAprism, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Steprāns.(a) There exists a family ℱ of less than continuummany functions from ℝ to ℝ such that ℝ2 is covered by functions from ℱ, in the sense that for every 〈x, y〉 ∈ ℝ2 there exists an f ∈ ℱ such that either f (x) = y or f (y) = x.(b) For every Borel function f : ℝ → ℝ there exists a family ℱ of less than continuum many “” functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of f.(c) For every n > 0 and a Dn function f: ℝ → ℝ there exists a family ℱ of less than continuum many Cn functions whose graphs cover the graph of f.We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevskiĭ.

scholarly journals Hypergraphs and proper forcing

2019 ◽  
Vol 19 (02) ◽  
pp. 1950007
Author(s):  
Jindřich Zapletal
Keyword(s):  
Polish Space ◽  
Countable Collection ◽  
The Family ◽  
Proper Forcing

Given a Polish space [Formula: see text] and a countable collection of analytic hypergraphs on [Formula: see text], I consider the [Formula: see text]-ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.

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