Aspects of Classical Descriptive Set Theory

TL;DR

This paper provides an overview of classical descriptive set theory, including fundamental concepts and results, and explores recent developments in invariant descriptive set theory, especially regarding equivalence relations between Polish metric spaces.

Contribution

It offers a comprehensive exposition of classical theory and introduces recent advances in invariant descriptive set theory related to Polish metric spaces.

Findings

Classical descriptive set theory concepts are systematically summarized.

The correspondence between classical and effective notions is highlighted.

Recent results on equivalence relations in Polish metric spaces are detailed.

Abstract

This report consists of two parts. The first part is a brief exposition of classical descriptive set theory. This part introduces some fundamental concepts, motivations and results from the classical theory and ends with a section on the important result of Addison that established the correspondence between classical and effective notions. The second part is about invariant descriptive set theory. It consists of a brief introduction to notions relevant to this branch of descriptive set theory and then describes in details some of the relatively recent results concerning equivalence relations between Polish metric spaces.