Preliminaries

This chapter reviews the concepts and results of classical and effective descriptive set theory that will be used in this book. Classical descriptive set theory was founded by Baire, Borel, Lebesgue, Luzin, Suslin, Sierpinski, and others in the first two decades of the twentieth century. The theory studies the descriptive complexity of sets of real numbers arising in ordinary mathematics, mostly in topology and analysis. The most striking achievements of this theory are the proofs of regularity properties of low-level definable sets of reals.