On Baire measurable colorings of group actions

2020 ◽  
pp. 1-28
Author(s):  
ANTON BERNSHTEYN
Keyword(s):  
Free Action ◽  
Descriptive Set Theory ◽  
Measurable Solution ◽  
Polish Spaces ◽  
Equivariant Maps ◽  
Generic Dynamics ◽  
Shift Action ◽  
Measurable Coloring ◽  
Descriptive Set ◽  
Dynamical Interpretation

The field of descriptive combinatorics investigates to what extent classical combinatorial results and techniques can be made topologically or measure-theoretically well behaved. This paper examines a class of coloring problems induced by actions of countable groups on Polish spaces, with the requirement that the desired coloring be Baire measurable. We show that the set of all such coloring problems that admit a Baire measurable solution for a particular free action $\unicode[STIX]{x1D6FC}$ is complete analytic (apart from the trivial situation when the orbit equivalence relation induced by $\unicode[STIX]{x1D6FC}$ is smooth on a comeager set); this result confirms the ‘hardness’ of finding a topologically well-behaved coloring. When $\unicode[STIX]{x1D6FC}$ is the shift action, we characterize the class of problems for which $\unicode[STIX]{x1D6FC}$ has a Baire measurable coloring in purely combinatorial terms; it turns out that closely related concepts have already been studied in graph theory with no relation to descriptive set theory. We remark that our framework permits a wholly dynamical interpretation (with colorings corresponding to equivariant maps to a given subshift), so this article can also be viewed as a contribution to generic dynamics.

COUNTABLE BOREL EQUIVALENCE RELATIONS

2002 ◽  
Vol 02 (01) ◽  
pp. 1-80 ◽  
Author(s):  
S. JACKSON ◽  
A. S. KECHRIS ◽  
A. LOUVEAU
Keyword(s):  
Set Theory ◽  
Equivalence Relations ◽  
Descriptive Set Theory ◽  
Polish Spaces ◽  
Borel Equivalence Relations ◽  
Universal Equivalence ◽  
Descriptive Set

This paper develops the foundations of the descriptive set theory of countable Borel equivalence relations on Polish spaces with particular emphasis on the study of hyperfinite, amenable, treeable and universal equivalence relations.

scholarly journals Borel and Hausdorff hierarchies in topological spaces of Choquet games and their effectivization

2014 ◽  
Vol 25 (7) ◽  
pp. 1490-1519 ◽  
Author(s):  
VERÓNICA BECHER ◽  
SERGE GRIGORIEFF
Keyword(s):  
Topological Spaces ◽  
Descriptive Set Theory ◽  
Stationary Strategy ◽  
Approximation Spaces ◽  
Polish Spaces ◽  
Proper Subclass ◽  
Continuous Domains ◽  
Algebraic Domains ◽  
Descriptive Set ◽  
Work Done

What parts of the classical descriptive set theory done in Polish spaces still hold for more general topological spaces, possibly T0 or T1, but not T2 (i.e. not Hausdorff)? This question has been addressed by Selivanov in a series of papers centred on algebraic domains. And recently it has been considered by de Brecht for quasi-Polish spaces, a framework that contains both countably based continuous domains and Polish spaces. In this paper, we present alternative unifying topological spaces, that we call approximation spaces. They are exactly the spaces for which player Nonempty has a stationary strategy in the Choquet game. A natural proper subclass of approximation spaces coincides with the class of quasi-Polish spaces. We study the Borel and Hausdorff difference hierarchies in approximation spaces, revisiting the work done for the other topological spaces. We also consider the problem of effectivization of these results.

scholarly journals Descriptive set Theory of Families of Small Sets

2007 ◽  
Vol 13 (4) ◽  
pp. 482-537 ◽  
Author(s):  
Étienne Matheron ◽  
Miroslav Zelený
Keyword(s):  
Set Theory ◽  
Descriptive Set Theory ◽  
Compact Sets ◽  
Polish Spaces ◽  
Survey Paper ◽  
Closed Sets ◽  
Descriptive Set

AbstractThis is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.

scholarly journals A comparison of concepts from computable analysis and effective descriptive set theory

2016 ◽  
Vol 27 (8) ◽  
pp. 1414-1436 ◽  
Author(s):  
VASSILIOS GREGORIADES ◽  
TAMÁS KISPÉTER ◽  
ARNO PAULY
Keyword(s):  
Set Theory ◽  
Metric Spaces ◽  
Computable Analysis ◽  
Descriptive Set Theory ◽  
Polish Spaces ◽  
Complete Metric Spaces ◽  
Cauchy Completion ◽  
Precise Relationship ◽  
Descriptive Set ◽  
Effective Descriptive Set Theory

Computable analysis and effective descriptive set theory are both concerned with complete metric spaces, functions between them and subsets thereof in an effective setting. The precise relationship of the various definitions used in the two disciplines has so far been neglected, a situation this paper is meant to remedy.As the role of the Cauchy completion is relevant for both effective approaches to Polish spaces, we consider the interplay of effectivity and completion in some more detail.

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.

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.

Two Applications of Inner Model Theory to the Study of Sets

1996 ◽  
Vol 2 (1) ◽  
pp. 94-107 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Los Angeles ◽  
Set Theory ◽  
Model Theory ◽  
Large Cardinals ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Definable Sets ◽  
Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

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