An overview of descriptive set theory

Abstract

Understanding the structure of sets of reals is one of the fundamental problems of set theory. Of course, an arbitrary subset of the continuum is immensely complicated, and exhibits in general no regularities. For example it could be non(Lebesgue)-measurable or fail to have the property of Baire. So from the early days of set theory, about a hundred years ago, mathematicians have concentrated on the study of various classes of definable sets of reals, wide enough to include the usual sets appearing in analysis, topology, etc., but also restricted to possibly exhibit a regular behaviour. The study of such classes of definable sets is the purpose of descriptive set theory. Indeed, to a large extent descriptive set theory is mainly concerned with the structure of the projective sets of reals, which we proceed to define.

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