Classical Descriptive Set Theory

Alexander S. Kechris (auth.)0387943749, 9780387943749, 3540943749

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory.
This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.

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Table of contents :
Front Matter….Pages i-xviii
Topological and Metric Spaces….Pages 1-4
Trees….Pages 5-12
Polish Spaces….Pages 13-17
Compact Metrizable Spaces….Pages 18-28
Locally Compact Spaces….Pages 29-30
Perfect Polish Spaces….Pages 31-34
Zero-dimensional Spaces….Pages 35-40
Baire Category….Pages 41-57
Polish Groups….Pages 58-64
Measurable Spaces and Functions….Pages 65-67
Borel Sets and Functions….Pages 68-72
Standard Borel Spaces….Pages 73-81
Borel Sets as Clopen Sets….Pages 82-84
Analytic Sets and the Separation Theorem….Pages 85-88
Borel Injections and Isomorphisms….Pages 89-93
Borel Sets and Baire Category….Pages 94-102
Borel Sets and Measures….Pages 103-119
Uniformization Theorems….Pages 120-128
Partition Theorems….Pages 129-136
Borel Determinacy….Pages 137-148
Games People Play….Pages 149-166
The Borel Hierarchy….Pages 167-178
Some Examples….Pages 179-189
The Baire Hierarchy….Pages 190-195
Representations of Analytic Sets….Pages 196-204
Universal and Complete Sets….Pages 205-208
Examples….Pages 209-216
Separation Theorems….Pages 217-225
Regularity Properties….Pages 226-233
Capacities….Pages 234-238
Analytic Well-founded Relations….Pages 239-241
Review….Pages 242-244
Examples….Pages 245-266
Co-Analytic Ranks….Pages 267-280
Rank Theory….Pages 281-298
Scales and Uniformization….Pages 299-312
The Projective Hierarchy….Pages 313-321
Projective Determinacy….Pages 322-326
The Periodicity Theorems….Pages 327-345
Epilogue….Pages 346-347
Back Matter….Pages 349-404