# New Directions in Descriptive Set Theory

@article{Kechris1999NewDI, title={New Directions in Descriptive Set Theory}, author={Alexander S. Kechris}, journal={Bulletin of Symbolic Logic}, year={1999}, volume={5}, pages={161 - 174} }

§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 ℝ n , ℂ n , (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…

## 64 Citations

### Linear algebraic groups and countable Borel equivalence relations

- Mathematics
- 2000

This paper is a contribution to the study of Borel equivalence relations on standard
Borel spaces (i.e., Polish spaces equipped with their Borel structure). In
mathematics one often deals with…

### Borel and analytic sets in locales.

- Mathematics
- 2020

We systematically develop analogs of basic concepts from classical descriptive set theory in the context of pointless topology. Our starting point is to take the elements of the free complete Boolean…

### Polish metric spaces: their classification and isometry groups

- MathematicsBull. Symb. Log.
- 2001

In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a…

### 2 8 Ja n 20 00 How Many Turing Degrees are There ?

- Mathematics
- 2000

A Borel equivalence relation on a Polish space is countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the…

### Rigidity Theorems For Actions Of Product Groups And Countable Borel Equivalence Relations

- Mathematics
- 2005

This Memoir is both a contribution to the theory of Borel equivalence relations,
considered up to Borel reducibility, and measure preserving group actions
considered up to orbit equivalence. Here E…

### On the classification of Polish metric spaces up to isometry

- Mathematics
- 2003

We study the classification problem of Polish metric spaces up to isometry and
the isometry groups of Polish metric spaces. In the framework of the descriptive
set theory of definable equivalence…

### Degree spectra and computable dimensions in algebraic structures

- MathematicsAnn. Pure Appl. Log.
- 2002

### Borel and Countably Determined Reducibility in Nonstandard Domain

- Mathematics
- 2002

Abstract. We consider, in a nonstandard domain, reducibility of equivalence relations in terms of the Borel reducibility ≤B and the countably determined (CD, for brevity) reducibility ≤CD. This…

### Uniformity, universality, and computability theory

- MathematicsJ. Math. Log.
- 2017

A number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations are proved, including the existence of refinements of Martin's ultrafilter on Turing invariant Borel sets to the invariant borel sets of equivalence Relations that are much finer than Turing equivalence.

## References

SHOWING 1-10 OF 37 REFERENCES

### Descriptive Set Theory

- Mathematics
- 1980

Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary…

### Analytic Ideals

- MathematicsBulletin of Symbolic Logic
- 1996

§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of…

### New Dichotomies for Borel Equivalence Relations

- MathematicsBulletin of Symbolic Logic
- 1997

Two new dichotomy theorems for Borel equivalence relations are announced and presented in context by giving an overview of related recent developments.

### The Descriptive Set Theory of Polish Group Actions: BETTER TOPOLOGIES

- Mathematics
- 1996

In this book the authors present their research into the foundations of the theory of Polish groups and the associated orbit equivalence relations. The particular case of locally compact groups has…

### The structure of hy-per nite Borel equivalence relations

- Mathematics
- 1994

We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not…

### The classification of hypersmooth Borel equivalence relations

- Mathematics
- 1997

This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which…

### Ergodic theory of amenable group actions. I: The Rohlin lemma

- Mathematics
- 1980

Classically, ergodic theory began with the study of flows or actions of R. Later, for technical reasons, much of the theory was first developed for actions of Z. More recently, there has been…

### Noncommutative Geometry

- Mathematics
- 1997

Noncommutative Spaces It was noticed a long time ago that various properties of sets of points can be restated in terms of properties of certain commutative rings of functions over those sets. In…

### Classical descriptive set theory

- Mathematics
- 1987

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the…

### Borel Equivalence Relations Induced by Actions of the Symmetric Group

- MathematicsAnn. Pure Appl. Log.
- 1998