New Directions in Descriptive Set Theory

Abstract

§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 calledPolish spaces. Typical examples are ℝn, ℂn, (separable) Hilbert space and more generally all separable Banach spaces, theCantor space2ℕ, theBaire spaceℕℕ, the infinite symmetric groupS∞, 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 theBorelsets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the correspondingBorel hierarchy(sets). After this come theprojective sets, obtained by starting with the Borel sets and closing under the operations of complementation and projection, and the correspondingprojective 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., theBorel(),analytic() andcoanalytic() sets.