Boolean operations, Borel sets, and Hausdorff's question

1996 ◽  
Vol 61 (4) ◽  
pp. 1287-1304
Author(s):  
Abhijit Dasgupta
Keyword(s):  
Polish Space ◽  
Descriptive Set Theory ◽  
Boolean Operation ◽  
Boolean Operations ◽  
Borel Sets ◽  
Systematic Fashion ◽  
Borel Hierarchy ◽  
The Difference ◽  
Projective Hierarchy ◽  
Descriptive Set

The study of infinitary Boolean operations was undertaken by the early researchers of descriptive set theory soon after Suslin's discovery of the important operation. The first attempt to lay down their theory in a systematic fashion was the work of Kantorovich and Livenson [5], where they call these the analytical operations. Earlier, Hausdorff had introduced the δs operations — essentially same as the monotoneω-ary Boolean operations, and Kolmogorov, independently of Hausdorff, had discovered the same objects, which were used in his study of the R operator.The ω-ary Boolean operations turned out to be closely related to most of the classical hierarchies over a fixed Polish space X, including, e. g., the Borel hierarchy (), the difference hierarchies of Hausdorff (Dη()), the C-hierarchy (Cξ) of Selivanovski, and the projective hierarchy (): for each of these hierarchies, every level can be expressed as the range of an ω-ary Boolean operation applied to all possible sequences of open subsets of X. In the terminology of Dougherty [3], every level is “open-ω-Boolean” (if and are collections of subsets of X and I is any set, is said to be -I-Boolean if there exists an I-ary Boolean operation Φ such that = Φ, i. e. is the range of Φ restricted to all possible I-sequences of sets from ). If in addition, the space X has a basis consisting of clopen sets, then the levels of the above hierarchies are also “clopen-ω-Boolean.”

scholarly journals New Directions in Descriptive Set Theory

1999 ◽  
Vol 5 (2) ◽  
pp. 161-174 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Hilbert Space ◽  
Set Theory ◽  
Descriptive Set Theory ◽  
Borel Sets ◽  
Polish Spaces ◽  
Definable Sets ◽  
Metrizable Spaces ◽  
In The Beginning ◽  
Projective Hierarchy ◽  
Descriptive Set

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are ℝn, ℂn, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2ℕ, the Baire space ℕℕ, the infinite symmetric group S∞, the unitary group (of the Hilbert space), the group of measure preserving transformations of the unit interval, etc.In this theory sets are classified in hierarchies according to the complexity of their definitions and the structure of sets in each level of these hierarchies is systematically analyzed. In the beginning we have the Borel sets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the corresponding Borel hierarchy ( sets). After this come the projective sets, obtained by starting with the Borel sets and closing under the operations of complementation and projection, and the corresponding projective hierarchy ( sets).There are also transfinite extensions of the projective hierarchy and even much more complex definable sets studied in descriptive set theory, but I will restrict myself here to Borel and projective sets, in fact just those at the first level of the projective hierarchy, i.e., the Borel (), analytic () and coanalytic () sets.

Optimal Proofs of Determinacy

1995 ◽  
Vol 1 (3) ◽  
pp. 327-339 ◽  
Author(s):  
Itay Neeman
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
The Other ◽  
Descriptive Set Theory ◽  
Transfer Theorem ◽  
Axiom Of Determinacy ◽  
Structural Connection ◽  
Projective Hierarchy ◽  
Descriptive Set

In this paper I shall present a method for proving determinacy from large cardinals which, in many cases, seems to yield optimal results. One of the main applications extends theorems of Martin, Steel and Woodin about determinacy within the projective hierarchy. The method can also be used to give a new proof of Woodin's theorem about determinacy in L(ℝ).The reason we look for optimal determinacy proofs is not only vanity. Such proofs serve to tighten the connection between large cardinals and descriptive set theory, letting us bring our knowledge of one subject to bear on the other, and thus increasing our understanding of both. A classic example of this is the Harrington-Martin proof that -determinacy implies -determinacy. This is an example of a transfer theorem, which assumes a certain determinacy hypothesis and proves a stronger one. While the statement of the theorem makes no mention of large cardinals, its proof goes through 0#, first proving that-determinacy ⇒ 0# exists,and then that0# exists ⇒ -determinacyMore recent examples of the connection between large cardinals and descriptive set theory include Steel's proof thatADL(ℝ) ⇒ HODL(ℝ) ⊨ GCH,see [9], and several results of Woodin about models of AD+, a strengthening of the axiom of determinacy AD which Woodin has introduced. These proofs not only use large cardinals, but also reveal a deep, structural connection between descriptive set theoretic notions and notions related to large cardinals.

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.

scholarly journals Towards a descriptive theory of cb0-spaces

2016 ◽  
Vol 27 (8) ◽  
pp. 1553-1580 ◽  
Author(s):  
VICTOR SELIVANOV
Keyword(s):  
Set Theory ◽  
Descriptive Set Theory ◽  
Difference Hierarchy ◽  
Descriptive Theory ◽  
Wadge Hierarchy ◽  
The Difference ◽  
Descriptive Set

The paper tries to extend some results of the classical Descriptive Set Theory to as many countably basedT0-spaces (cb0-spaces) as possible. Along with extending some central facts about Borel, Luzin and Hausdorff hierarchies of sets we also consider the more general case ofk-partitions. In particular, we investigate the difference hierarchy ofk-partitions and the fine hierarchy closely related to the Wadge hierarchy.

Some results about Borel sets in descriptive set theory of hyperfinite sets

1990 ◽  
Vol 55 (2) ◽  
pp. 604-614 ◽  
Author(s):  
Boško Živaljević
Keyword(s):  
Set Theory ◽  
Descriptive Set Theory ◽  
Counting Measure ◽  
Borel Sets ◽  
Similar Characterization ◽  
Borel Functions ◽  
Remarkable Result ◽  
Descriptive Set ◽  
The Given ◽  
Internal Sets

A remarkable result of Henson and Ross [HR] states that if a function whose graph is Souslin in the product of two hyperfinite sets in an ℵ1 saturated nonstandard universe possesses a certain nice property (capacity) then there exists an internal subfunction of the given one possessing the same property. In particular, they prove that every 1-1 Souslin function preserves any internal counting measure, and show that every two internal sets A and B with ∣A∣/∣B∣ ≈ 1 are Borel bijective. As a supplement to the last-mentioned result of Henson and Ross, Keisler, Kunen, Miller and Leth showed [KKML] that two internal sets A and B are bijective by a countably determined bijection if and only if ∣A∣/∣B∣ is finite and not infinitesimal.In this paper we first show that injective Borel functions map Borel sets into Borel sets, a fact well known in classical descriptive set theory. Then, we extend the result of Henson and Ross concerning the Borel bijectivity of internal sets whose quotient of cardinalities is infinitely closed to 1. We prove that two Borel sets, to which we may assign a counting measure not equal to 0 or ∞, are Borel bijective if and only if they have the same counting measure ≠0, ∞. This, together with the similar characterization for Souslin and measurable countably determined sets, extends the above-mentioned results from [HR] and [KKML].

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.

scholarly journals Turing degrees in Polish spaces and decomposability of Borel functions

2020 ◽  
Vol 21 (01) ◽  
pp. 2050021
Author(s):  
Vassilios Gregoriades ◽  
Takayuki Kihara ◽  
Keng Meng Ng
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Recursion Theory ◽  
Descriptive Set Theory ◽  
Analytic Subset ◽  
Polish Spaces ◽  
Negative Results ◽  
Borel Functions ◽  
Important Open Problem ◽  
Descriptive Set

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture.

scholarly journals GREY SUBSETS OF POLISH SPACES

2015 ◽  
Vol 80 (4) ◽  
pp. 1379-1397 ◽  
Author(s):  
ITAÏ BEN YAACOV ◽  
JULIEN MELLERAY
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Descriptive Set Theory ◽  
Polish Spaces ◽  
Small Index Property ◽  
Small Index ◽  
Descriptive Set

AbstractWe develop the basics of an analogue of descriptive set theory for functions on a Polish space X. We use this to define a version of the small index property in the context of Polish topometric groups, and show that Polish topometric groups with ample generics have this property. We also extend classical theorems of Effros and Hausdorff to the topometric context.

Topological Manifolds and Polish Spaces

2021 ◽  
Author(s):  
Yu-Lin Chou
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Open Cover ◽  
Descriptive Set Theory ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Polish Spaces ◽  
Manifold With Boundary ◽  
Topological Manifolds ◽  
Descriptive Set

We give,as a preliminary result, some topological characterizations of locally compact second-countable Hausdorff spaces. Then we show that a topological manifold, with boundary or not,is precisely a Polish space with a coordinate open cover; this connects geometry with descriptive set theory.

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