Classical Descriptive Set TheoryDescriptive 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. |
Contents
| 5 | |
Polish Groups | 9 |
Polish Spaces | 13 |
CHAPTER II | 22 |
Borel Sets and Functions | 68 |
Borel Sets as Clopen Sets | 82 |
Borel Injections and Isomorphisms | 89 |
Borel Sets and Measures | 103 |
Separation Theorems | 217 |
Regularity Properties | 226 |
Capacities | 234 |
CHAPTER IV | 242 |
CoAnalytic Ranks | 267 |
Rank Theory | 281 |
Standard Borel Spaces | 286 |
Scales and Uniformization | 299 |
Uniformization Theorems | 120 |
Partition Theorems | 129 |
Borel Determinacy | 137 |
Games People Play | 149 |
The Borel Hierarchy | 167 |
Some Examples | 179 |
The Baire Hierarchy | 190 |
CHAPTER III | 196 |
Universal and Complete Sets | 205 |
CHAPTER V | 313 |
Projective Determinacy | 322 |
Epilogue | 346 |
On Logical Notation | 353 |
| 369 | |
Symbols and Abbreviations | 381 |
| 387 | |
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Common terms and phrases
algebra analytic sets assume B₁ Baire Banach space bijection 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 continuous preimages contradiction converges define denote Determinacy equivalence relation Exercise finite fn(x function f given Gs set homeomorphic II-complete II-rank infinite Kechris Lemma length(s Let X,Y Lusin meager measurable space metrizable space nonempty open sets notation open nbhd open sets ordinal player plays pointwise Polish space projx Proof prove quasistrategy rank recursion separable Banach spaces sequence set AC set theory sets in Polish Show space and AC standard Borel space strategy in G topological space U₁ uniformization unique w₁ Wadge well-founded winning strategy wins iff zero-dimensional