WADGE HIERARCHY OF DIFFERENCES OF CO-ANALYTIC SETS

2016 ◽  
Vol 81 (1) ◽  
pp. 201-215 ◽  
Author(s):  
KEVIN FOURNIER
Keyword(s):  
Baire Space ◽  
Borel Sets ◽  
Wadge Hierarchy ◽  
Analytic Sets ◽  
Image Position

AbstractWe begin the fine analysis of nonBorel pointclasses. Working in ZFC + DET$\left( {_1^1 } \right)$, we describe the Wadge hierarchy of the class of increasing differences of co-analytic subsets of the Baire space by extending results obtained by Louveau ([5]) for the Borel sets.

Sequential discreteness and clopen-I-Boolean classes

1987 ◽  
Vol 52 (1) ◽  
pp. 232-242
Author(s):  
Randall Dougherty
Keyword(s):  
Final Section ◽  
Baire Space ◽  
Boolean Operations ◽  
Index Set ◽  
Borel Sets ◽  
Borel Hierarchy ◽  
Wadge Reducibility ◽  
Analytic Sets ◽  
Converse Results

Kantorovich and Livenson [6] initiated the study of infinitary Boolean operations applied to the subsets of the Baire space and related spaces. It turns out that a number of interesting collections of subsets of the Baire space, such as the collection of Borel sets of a given type (e.g. the Fσ sets) or the collection of analytic sets, can be expressed as the range of an ω-ary Boolean operation applied to all possible ω-sequences of clopen sets. (Such collections are called clopen-ω-Boolean.) More recently, the ranges of I-ary Boolean operations for uncountable I have been considered; specific questions include whether the collection of Borel sets, or the collection of sets at finite levels in the Borel hierarchy, is clopen-I-Boolean.The main purpose of this paper is to give a characterization of those collections of subsets of the Baire space (or similar spaces) that are clopen-I-Boolean for some I. The Baire space version can be stated as follows: a collection of subsets of the Baire space is clopen-I-Boolean for some I iff it is nonempty and closed downward and σ-directed upward under Wadge reducibility, and in this case we may take I = ω2. The basic method of proof is to use discrete subsets of spaces of the form K2 to put a number of smaller clopen-I-Boolean classes together to form a large one. The final section of the paper gives converse results indicating that, at least in some cases, ω2 cannot be replaced by a smaller index set.

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).

sets of reals

1997 ◽  
Vol 62 (4) ◽  
pp. 1379-1428 ◽  
Author(s):  
Joan Bagaria ◽  
W. Hugh Woodin
Keyword(s):  
Set Theory ◽  
Real Numbers ◽  
Borel Sets ◽  
Definable Sets ◽  
Analytic Sets ◽  
Basic Facts ◽  
Continuous Images

Some of the most striking results in modern set theory have emerged from the study of simply-definable sets of real numbers. Indeed, simple questions like: what are the posible cardinalities?, are they measurable?, do they have the property of Baire?, etc., cannot be answered in ZFC.When one restricts the attention to the analytic sets, i.e., the continuous images of Borel sets, then ZFC does provide an answer to these questions. But this is no longer true for the projective sets, i.e., all the sets of reals that can be obtained from the Borel sets by taking continuous images and complements. In this paper we shall concentrate on particular projective classes, the , and using forcing constructions we will produce models of ZFC where, for some n, all , sets have some specified property. For the definition and basic facts about the projective classes , and , as well as the Kleene (or lightface) classes , and , we refer the reader to Moschovakis [19].The first part of the paper is about measure and category. Early in this century, Luzin [16] and Luzin-Sierpiński [17] showed that all analytic (i.e., ) sets of reals are Lebesgue measurable and have the property of Baire.

scholarly journals THE GAME OPERATOR ACTING ON WADGE CLASSES OF BOREL SETS

2019 ◽  
Vol 84 (3) ◽  
pp. 1224-1239
Author(s):  
GABRIEL DEBS ◽  
JEAN SAINT RAYMOND
Keyword(s):  
Borel Sets ◽  
Substitution Property ◽  
Image Position

AbstractWe study the behavior of the game operator $$ on Wadge classes of Borel sets. In particular we prove that the classical Moschovakis results still hold in this setting. We also characterize Wadge classes ${\bf{\Gamma }}$ for which the class has the substitution property. An effective variation of these results shows that for all $1 \le \eta < \omega _1^{{\rm{CK}}}$ and $2 \le \xi < \omega _1^{{\rm{CK}}}$, is a Spector class while is not.

scholarly journals The Borel structure of iterates of continuous functions

1989 ◽  
Vol 32 (3) ◽  
pp. 483-494 ◽  
Author(s):  
Paul D. Humke ◽  
M. Laczkovich
Keyword(s):  
Banach Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Space Of Continuous Functions ◽  
Borel Structure ◽  
Image Position ◽  
Set Of Points ◽  
Positive Integers

Let C[0,1] be the Banach space of continuous functions defined on [0,1] and let C be the set of functions f∈C[0,1] mapping [0,1] into itself. If f∈C, fk will denote the kth iterate of f and we put Ck = {fk:f∈C;}. The set of increasing (≡ nondecreasing) and decreasing (≡ nonincreasing) functions in C will be denoted by ℐ and D, respectively. If a function f is defined on an interval I, we let C(f) denote the set of points at which f is locally constant, i.e.We let N denote the set of positive integers and NN denote the Baire space of sequences of positive integers.

scholarly journals Wadge-like reducibilities on arbitrary quasi-Polish spaces

2014 ◽  
Vol 25 (8) ◽  
pp. 1705-1754 ◽  
Author(s):  
LUCA MOTTO ROS ◽  
PHILIPP SCHLICHT ◽  
VICTOR SELIVANOV
Keyword(s):  
Real Line ◽  
Polish Space ◽  
Baire Space ◽  
Topological Spaces ◽  
Polish Spaces ◽  
The Real ◽  
Degree Structure ◽  
Wadge Hierarchy

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well ordered), but for many other natural nonzero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called Δ0α-reductions, and try to find for various natural topological spaces X the least ordinal αX such that for every αX ⩽ β < ω1 the degree-structure induced on X by the Δ0β-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that αX ⩽ ω for every quasi-Polish space X, that αX ⩽ 3 for quasi-Polish spaces of dimension ≠ ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.

Sufficient Statistics and Orthogonal Parameters

1962 ◽  
Vol 58 (2) ◽  
pp. 326-337 ◽  
Author(s):  
Ann F. S. Mitchell
Keyword(s):  
Probability Density Function ◽  
Finite Measure ◽  
Random Variable ◽  
Sample Space ◽  
Continuous Random Variable ◽  
Sufficient Statistics ◽  
Counting Measure ◽  
Borel Sets ◽  
Orthogonal Parameters ◽  
Image Position

Let be, for a set of n real continuous parameters the probability density function of a random variable x with respect to a σ-finite measure μ on a σ-algebra of subsets of the sample space . If x; is a continuous random variable, μ will be Lebesgue measure on the Borel sets of a Euclidean sample space and, if x is discrete, μ will be counting measure on the class of all sets of a countable sample space. The parameters αi are said to be orthogonal (Jeffreys (3), pp. 158,184) if .

Ergodic Averages for Weight Functions Moved by Non-Linear Transformations on ℝn

1995 ◽  
Vol 47 (4) ◽  
pp. 852-876
Author(s):  
David I. McIntosh
Keyword(s):  
Sufficient Conditions ◽  
Convergent Sequence ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Linear Transformations ◽  
Borel Sets ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Sets

AbstractLet ℝ+ denote the non-negative half of the real line, and let λ denote Lebesgue measure on the Borel sets of ℝn. A function φ: ℝn → ℝ+ is called a weight function if ʃℝn φ dλ = 1. Let (X, ℱ, μ) be a non-atomic, finite measure space, let ƒ: X → ℝ+, and suppose { Tν}ν∊ℝn is an ergodic, aperiodic ℝn-flow on X. We consider the weighted ergodic averages where is a sequence of weight functions. Sufficient as well as necessary and sufficient conditions for the pointwise, almost-everywhere convergence of are developed for a particular class of weight functions φk. Specifically, let {τk: ℝn → ℝn} be a sequence of measurable, non-singular maps with measurable, non-singular inverses such that the Radon-Nikodym derivatives dλ oτk /dλ and dλ oτk-1 / dλ are L∞ (ℝn), and such that τk and τ-1 map bounded sets to bounded sets. We examine convergence for the sequence where θk is an a.e.-convergent sequence of weight functions which are dominated by a fixed L1(ℝn) function with bounded support.

Hurewicz test sets for generalized separation and reduction

2007 ◽  
Vol 143 (2) ◽  
pp. 407-417
Author(s):  
TAMÁS MÁTRAI
Keyword(s):  
Analogous Result ◽  
Type Theorem ◽  
Borel Sets ◽  
Analytic Sets ◽  
Test Sets

AbstractWe prove a Hurewicz-type theorem for generalized separation: we present a method which allows us to test if for a sequence of Borel sets (Ai)i > ω satisfying $\bigcap_{i <\omega} A_{i} = \phis$ there is a sequence (Bi)i < ω of Π0ξ sets such that $A_{i} \ss B_{i}$ $(i <\omega)$ and $\bigcap_{i <\omega} B_{i} = \phis$ or not. We also prove an analogous result for generalized reduction. The results of the paper are motivated by a Hurewicz-type theorem of A. Louveau and J. Saint Raymond on ordinary separation of analytic sets.

On partitioning the infinite subsets of large cardinals

1984 ◽  
Vol 49 (2) ◽  
pp. 539-541 ◽  
Author(s):  
R. J. Watro
Keyword(s):  
Order Type ◽  
Large Cardinal ◽  
Infinite Cardinal ◽  
Product Topology ◽  
Weakly Compact ◽  
Ramsey Theorem ◽  
Borel Sets ◽  
Analytic Sets ◽  
The Axiom Of Choice ◽  
Usual Product

Let λ be an ordinal less than or equal to an infinite cardinal κ. For S ⊂ κ, [S]λ denotes the collection of all order type λ subsets of S. A set X ⊂ [κ]λ will be called Ramsey iff there exists p ∈ [κ]κ such that either [p]λ ⊂ X or [p]λ ∩ X = ∅. The set p is called homogeneous for X.The infinite Ramsey theorem implies that all subsets of [ω]n are Ramsey for n < ω. Using the axiom of choice, one can define a non-Ramsey subset of [ω]ω. In [GP], Galvin and Prikry showed that all Borel subsets of [ω]ω are Ramsey, where one topologizes [ω]ω as a subspace of Baire space. Silver [S] proved that analytic sets are Ramsey, and observed that this is best possible in ZFC.When κ > ω, the assertion that all subsets of [κ]n are Ramsey is a large cardinal hypothesis equivalent to κ being weakly compact (and strongly inaccessible). Again, is not possible in ZFC to have all subsets of [κ]ω Ramsey. The analogy to the Galvin-Prikry theorem mentioned above was established by Kleinberg, extending work by Kleinberg and Shore in [KS]. The set [κ]ω is given a topology as a subspace of κω, which has the usual product topology, κ taken as discrete. It was shown that all open subsets of [κ]ω are Ramsey iff κ is a Ramsey cardinal (that is, κ → (κ)<ω).In this note we examine the spaces [κ]λ for κ ≥ λ ≥ ω. We show that κ Ramsey implies all open subsets of [κ]λ are Ramsey for λ < κ, and that if κ is measurable, then all open subsets of [κ]κ are Ramsey. Let us remark here that we can with the same methods prove these results with “κ-Borel” in the place of “open”, where the κ-Borel sets are the smallest collection containing the opens and closed under complementation and intersections of length less than κ. Also, although here we consider just subsets of [κ]λ, it is no more difficult to show that partitions of [κ]λ into less than κ many κ-Borel sets have, under the appropriate hypothesis, size κ homogeneous sets.

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