Classical Descriptive Set TheoryDescriptive set theory is the area of mathematics concerned with the study of the structure of definable sets in Polish spaces. Beyond being a central part of contemporary set theory, the concepts and results of descriptive set theory are being used in diverse fields of mathematics, such as logic, combinatorics, topology, Banach space theory, real and harmonic analysis, potential theory, ergodic theory, operator algebras, and group representation theory. This book provides a basic first introduction to the subject at the beginning graduate level. It concentrates on the core classical aspects, but from a modern viewpoint, including many recent developments, like games and determinacy, and illustrates the general theory by numerous examples and applications to other areas of mathematics. The book, which is written in the style of informal lecture notes, consists of five chapters. The first contains the basic theory of Polish spaces and its standard tools, like Baire category. The second deals with the theory of Borel sets. Methods of infinite games figure prominently here as well as in subsequent chapters. The third chapter is devoted to the analytic sets and the fourth to the co-analytic sets, developing the machinery associated with ranks and scales. The final chapter gives an introduction to the projective sets, including the periodicity theorems. The book contains over four hundred exercises of varying degrees of difficulty. |
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algebra analytic sets assume B₁ Baire space Banach space bijection Boolean Borel function Borel isomorphism Borel measure Borel sets C-universal called Cantor set Choquet class of sets clearly clopen closed set closed subspace comeager compact metrizable compact sets Consider contains continuous function contradiction converges define denote Determinacy equivalence relation Exercise finite fn(x function f given Gs set homeomorphic II-complete II-rank infinite Lemma length(s Let X,Y Lusin meager measurable space metrizable space nonempty open sets notation open nbhd open sets pairwise disjoint player plays pointwise Polish group Polish space projx Proof prove quasistrategy recursion resp second countable separable Banach space sequence set AC Show Souslin scheme standard Borel space topological space U₁ uniformization unique V₁ w₁ Wadge well-founded winning strategy wins iff zero-dimensional