scholarly journals Borel sets and countable models

2011 ◽  
Vol 90 (104) ◽  
pp. 1-11
Author(s):  
Zarko Mijajlovic ◽  
Dragan Doder ◽  
Angelina Ilic-Stepic
Keyword(s):  
Polish Space ◽  
Borel Sets ◽  
Domino Tiling ◽  
Theoretical Assumptions ◽  
Countable Structure ◽  
Countable Models

We show that certain families of sets and functions related to a countable structure A are analytic subsets of a Polish space. Examples include sets of automorphisms, endomorphisms and congruences of A and sets of the combinatorial nature such as coloring of countable plain graphs and domino tiling of the plane. This implies, without any additional set-theoretical assumptions, i.e., in ZFC alone, that cardinality of every such uncountable set is 2?0.

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.”

OCCUPATION TIMES OF BROWNIAN SEGMENTS AND THE σ-FINITE WIENER MEASURE

2005 ◽  
Vol 08 (02) ◽  
pp. 199-213
Author(s):  
TOMASZ BOJDECKI ◽  
LUIS G. GOROSTIZA
Keyword(s):  
Brownian Motion ◽  
Polish Space ◽  
Wiener Measure ◽  
Asymptotic Result ◽  
Main Role ◽  
Occupation Times ◽  
Borel Sets ◽  
New Information ◽  
Strengthened Form ◽  
Consecutive Time

We give an asymptotic result for the occupation of Borel sets of functions by the segments of recurrent Brownian motion on consecutive time intervals [n, n +1], n =0, 1, 2, …. This result provides new information on the behavior of Brownian motion, which is illustrated by examples. A formulation in terms of weak convergence of random measures on Polish space is also given. The proof is based on (a strengthened form of) the Darling–Kac occupation time theorem for Markov chains, and our result can be viewed as a "trajectorial" extension of that theorem. The main role in the occupation limit for Brownian segments is played by the σ-finite Wiener measure, which first appeared in a different context. An extension for segments of symmetric α-stable Lévy processes is also given.

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.

Π11 Borel sets

1989 ◽  
Vol 54 (3) ◽  
pp. 915-920 ◽  
Author(s):  
Alexander S. Kechris ◽  
David Marker ◽  
Ramez L. Sami
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Final Section ◽  
Borel Sets ◽  
Abstract Setting ◽  
Recursive Theory ◽  
Scott Rank ◽  
Countable Models

The results in this paper were motivated by the following question of Sacks. Suppose T is a recursive theory with countably many countable models. What can you say about the least ordinal α such that all models of T have Scott rank below α? If Martin's conjecture is true for T then α ≤ ω · 2.Our goal was to look at this problem in a more abstract setting. Let E be a equivalence relation on ωω with countably many classes each of which is Borel. What can you say about the least α such that each equivalence class is ? This problem is closely related to the following question. Suppose X ⊆ ωω is and Borel. What can you say about the least α such that X is ?In §1 we answer these questions in ZFC. In §2 we give more informative answers under the added assumptions V = L or -determinacy. The final section contains related results on the separation of sets by Borel sets.Our notation is standard. The reader may consult Moschovakis [5] for undefined terms.Some of these results were proved first by Sami and rediscovered by Kechris and Marker.

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 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.

A quasi-order on continuous functions

2013 ◽  
Vol 78 (2) ◽  
pp. 633-648 ◽  
Author(s):  
Raphaël Carroy
Keyword(s):  
Polish Space ◽  
Continuous Functions ◽  
Borel Sets ◽  
Borel Functions ◽  
Quasi Order

AbstractWe define a quasi-order on Borel functions from a zero-dimensional Polish space into another that both refines the order induced by the Baire hierarchy of functions and generalises the embeddability order on Borel sets. We study the properties of this quasi-order on continuous functions, and we prove that the closed subsets of a zero-dimensional Polish space are well-quasi-ordered by bi-continuous embeddability.

Louveau's theorem for the descriptive set theory of internal sets

1997 ◽  
Vol 62 (2) ◽  
pp. 595-607
Author(s):  
Kenneth Schilling ◽  
Boško Živaljević
Keyword(s):  
Polish Space ◽  
Borel Subset ◽  
Measurable Space ◽  
Borel Sets ◽  
Polish Spaces ◽  
Borel Hierarchy ◽  
Open Questions ◽  
Vertical Sections ◽  
Almost All ◽  
Internal Sets

AbstractWe give positive answers to two open questions from [15]. (1) For every set C countably determined over , if C is then it must be over , and (2) every Borel subset of the product of two internal sets X and Y all of whose vertical sections are can be represented as an intersection (union) of Borel sets with vertical sections of lower Borel rank. We in fact show that (2) is a consequence of the analogous result in the case when X is a measurable space and Y a complete separable metric space (Polish space) which was proved by A. Louveau and that (1) is equivalent to the property shared by the inverse standard part map in Polish spaces of preserving almost all levels of the Borel hierarchy.

Behavioral and emotional differences between children of divorce and children from intact families: Clinical significance and mediating processes *This study is part of a dissertation () supported from the German research community.

2002 ◽  
Vol 61 (1) ◽  
pp. 5-14 ◽  
Author(s):  
Andreas Schick
Keyword(s):  
Self Esteem ◽  
Research Community ◽  
Children Of Divorce ◽  
Group Differences ◽  
Significant Group ◽  
Intact Families ◽  
Divorce Adjustment ◽  
Theoretical Assumptions ◽  
German Research ◽  
Clinically Significant

The following study is based on a sample of 241 9-13-year-old children (66 children from divorced parents, 175 children from non divorced parents). They were examined for differences regarding anxiety, self-esteem, different areas of competence, and degree of behavior problems. With a focus on the children’s experiences, the clinically significant differences were examined. Clinically significant differences, revealing more negative outcomes for the children of divorce, were only found for social anxiety and unstable performance. The frequency of clinical significant differences was independent of the length of time the parents had been separated. The perceived destructiveness of conflict between the parents one of four facets of interparental conflict in this study functioned as a central mediator of the statistically significant group differences. The children’s perception of the father’s social support was a less reliable indicator of variance. Further studies should try to make underlying theoretical assumptions about the effects of divorce more explicit, to distinguish clearly between mediating variables, and to investigate them with respect to specific divorce adjustment indicators.

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