Classical descriptive set theory:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New York [u.a.]
Springer
1995
|
| Schriftenreihe: | Graduate texts in mathematics
156 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006557093&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XVIII, 402 S. graph. Darst. |
| ISBN: | 3540943749 0387943749 |
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| 100 | 1 | |a Kechris, Alexander S. |d 1946- |e Verfasser |0 (DE-588)109300025 |4 aut | |
| 245 | 1 | 0 | |a Classical descriptive set theory |c Alexander S. Kechris |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1995 | |
| 300 | |a XVIII, 402 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 156 | |
| 650 | 4 | |a Ensembles, Théorie des | |
| 650 | 7 | |a Ensembles, théorie des |2 ram | |
| 650 | 7 | |a Verzamelingen (wiskunde) |2 gtt | |
| 650 | 4 | |a Set theory | |
| 650 | 0 | 7 | |a Deskriptive Mengenlehre |0 (DE-588)4149180-4 |2 gnd |9 rswk-swf |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-006557093 | |
Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 MAT 040f 2001 A 20393 |
|---|---|
| DE-BY-TUM_katkey | 639407 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040010368966 |
| _version_ | 1821937882520092672 |
| adam_text | Contents
Preface vii
Introduction xv
About This Book xvii
Chapter I
Polish Spaces 1
1. Topological and Metric Spaces 1
l.A Topological Spaces 1
l.B Metric Spaces 2
2. Trees 5
2.A Ba.sk Concepts 5
2.B Trees and Closed Sets 7
2.C Trees on Products 9
2.D Leftmost Branches 9
2.E Well founded Trees and Ranks 10
2.F The Well founded Part of a Tree 11
2.G The Kleene Brouwer Ordering 11
3. Polish Spaces 13
3.A Definitions and Examples 13
3.B Extensions of Continuous Functions and Homeomorphisms 15
3.C Polish Subspaces of Polish Spaces 17
4. Compact Metrizable Spaces 18
4.A Basic Facts 18
4.B Examples 19
x Contents
4.C A Universality Property of the Hilbert Cube . 22
4.D Continuous Images of the Cantor Space 23
4.E The Space of Continuous Functions on a Compact Space 24
4.F The Hyperspace of Compact Sets 24
5. Locally Compact Spaces 29
6. Perfect Polish Spaces 31
6.A Embedding the Cantor Space in Perfect Polish Spaces 31
6.B The Cantor Bendixson Theorem 32
6.C Cantor Bendixson Derivatives and Ranks 33
7. Zero dimensional Spaces 35
7.A Basic Facts 35
7.B A Topological Characterization of the Cantor Space 35
7.C A Topological Characterization of the Baire Space 36
7.D Zero dimensional Spaces as Subspaces of the Baire Space 38
7.E Polish Spaces as Continuous Images of the Baire Space 38
7.F Closed Subsets Homeomorphic to the Baire Space 39
8. Baire Category 41
8.A Meager Sets 41
8.B Baire Spaces 41
8.C Choquet Games and Spaces 43
8.D Strong Choquet Games and Spaces 44
8.E A Characterization of Polish Spaces 45
8.F Sets with the Baire Property 47
8.G Localization 48
8.H The Banach Mazur Game 51
8.1 Baire Measurable Functions 52
8.J Category Quantifiers 53
8.K The Kuratowski Ulam Theorem 53
8.L Some Applications 55
8.M Separate and Joint Continuity 56
9. Polish Groups 58
9.A Metrizable and Polish Groups 58
9.B Examples of Polish Groups 58
9.C Basic Facts about Baire Groups and Their Actions 60
9.D Universal Polish Groups 63
Chapter II
Borel Sets 65
10. Measurable Spaces and Functions 65
10.A Sigma Algebras and Their Generators 65
10.B Measurable Spaces and Functions 66
11. Borel Sets and Functions 68
11. A Borel Sets in Topological Spaces 68
ll.B The Borel Hierarchy 68
ll.C Borel Functions 70
12. Standard Borel Spaces 73
12.A Borel Sets and Functions in Separable Metrizable Spaces 73
12.B Standard Borel Spaces 74
Contents xi
12. C The Effros Borel Space 75
12.D An Application to Selectors 77
12.E Further Examples 78
12.F Standard Borel Groups 80
13. Borel Sets as Clopen Sets 82
13.A Turning Borel into Clopen Sets 82
13.B Other Representations of Borel Sets 83
13.C Turning Borel into Continuous Functions 84
14. Analytic Sets and the Separation Theorem 85
14.A Basic Facts about Analytic Sets 85
14.B The Lusin Separation Theorem 87
14.C Souslin s Theorem 87
15. Borel Injections and Isomorphisms 89
15. A Borel Injective Images of Borel Sets 89
15.B The Isomorphism Theorem 90
15.C Homomorphisms of Sigma Algebras Induced by Point Maps 91
15.D Some Applications to Group Actions 92
16. Borel Sets and Baire Category 94
16. A Borel Definability of Category Notions 94
16.B The Vaught Transforms 95
16.C Connections with Model Theory 96
16.D Connections with Cohen s Forcing Method 99
17. Borel Sets and Measures 103
17. A General Facts on Measures 103
17.B Borel Measures 105
17.C Regularity and Tightness of Measures 107
17.D Lusin s Theorem on Measurable Functions 108
17.E The Space of Probability Borel Measures 109
17.F The Isomorphism Theorem for Measures 116
18. Uniformization Theorems 120
18.A The Jankov, von Neumann Uniformization Theorem 120
18.B Large Section Uniformization Results 122
18.C Small Section Uniformization Results 123
18.D Selectors and Transversals 128
19. Partition Theorems 129
19.A Partitions with a Comeager or Non meager Piece 129
19.B A Ramsey Theorem for Polish Spaces 130
19.C The Galvin Prikry Theorem 132
19.D Ramsey Sets and the Ellentuck Topology 132
19.E An Application to Banach Space Theory 134
20. Borel Determinacy I37
20.A Infinite Games 137
20.B Determinacy of Closed Games 138
20.C Borel Determinacy 140
20.D Game Quantifiers 147
21. Games People Play 149
21. A The * Games 149
21.B Unfolding 150
21.C The Banach Mazur or ** Games 151
xii Contents
21.D The General Unfolded Banach Mazur Games 153
21.E Wadge Games 156
21.F Separation Games and Hurewicz s Theorem 160
21.G Turing Degrees 164
22. The Borel Hierarchy 167
22.A Universal Sets 167
22.B The Borel versus the Wadge Hierarchy 169
22.C Structural Properties 170
22.D Additional Results 173
22.E The Difference Hierarchy 175
23. Some Examples 179
23.A Combinatorial Examples 179
23.B Classes of Compact Sets 181
23.C Sequence Spaces 182
23.D Classes of Continuous Functions 182
23.E Uniformly Convergent Sequences 185
23.F Some Universal Sets 185
23.G Further Examples 188
24. The Baire Hierarchy 190
24. A The Baire Classes of Functions 190
24.B Functions of Baire Class 1 192
Chapter III
Analytic Sets 196
25. Representations of Analytic Sets 196
25.A Review 196
25.B Analytic Sets in the Baire Space 197
25.C The Souslin Operation 198
25.D Wellordered Unions and Intersections of Borel Sets 201
25.E Analytic Sets as Open Sets in Strong Choquet Spaces 202
26. Universal and Complete Sets 205
26.A Universal Analytic Sets 205
26.B Analytic Determinacy 205
26.C Complete Analytic Sets 206
26.D Classification up to Borel Isomorphism 207
27. Examples 209
27. A The Class of Ill founded Trees 209
27.B Classes of Closed Sets 209
27.C Classes of Structures in Model Theory 212
27.D Isomorphism 213
27.E Some Universal Sets 214
27.F Miscellanea 215
28. Separation Theorems 217
28.A The Lusin Separation Theorem Revisited 217
28.B The Novikov Separation Theorem 219
28.C Borel Sets with Open or Closed Sections 220
28.D Some Special Separation Theorems 221
28.E ; Hurewicz Type Separation Theorems 224
Contents xiii
29. Regularity Properties 226
29.A The Perfect Set Property 226
29.B Measure, Category, and Ramsey 226
29.C A Closure Property for the Souslin Operation 227
29.D The Class of C Sets 230
29.E Analyticity of Largeness Conditions on Analytic Sets 230
30. Capacities 234
30.A The Basic Concept 234
30.B Examples 234
30. C The Choquet Capacitability Theorem 237
31. Analytic Well founded Relations 239
31.A Bounds on Ranks of Analytic Well founded Relations 239
31.B The Kunen Martin Theorem 241
Chapter IV
Co Analytic Sets 242
32. Review 242
32.A Basic Facts 242
32.B Representations of Co Analytic Sets 243
32.C Regularity Properties 244
33. Examples 245
33.A Well founded Trees and Wellorderings 245
33.B Classes of Closed Sets 245
33.C Sigma Ideals of Compact Sets 246
33.D Differentiable Functions 248
33.E Everywhere Convergence 251
33.F Parametrizing Baire Class 1 Functions 252
33.G A Method for Proving Completeness 253
33.H Singular Functions 254
33.1 Topological Examples 255
33. J Homeomorphisms of Compact Spaces 257
33.K Classes of Separable Banach Spaces 262
33.L Other Examples 266
34. Co Analytic Ranks 267
34.A Ranks and Prewellorderings 267
34.B Ranked Classes 267
34.C Co Analytic Ranks 268
34.D Derivatives 270
34.E Co Analytic Ranks Associated with Borel Derivatives 272
34.F Examples 275
35. Rank Theory 281
35.A Basic Properties of Ranked Classes 281
35.B Parametrizing Bi Analytic and Borel Sets 283
35.C Reflection Theorems 285
35.D Boundedness Properties of Ranks 288
35. E The Rank Method 290
35.F The Strategic Uniformization Theorem 291
35.G Co Analytic Families of Closed Sets and Their Sigma Ideals 292
xiv Contents
35.H Borel Sets with Fa and Ka Sections 296
36. Scales and Uniformization 299
36.A Kappa Souslin Sets 299
36.B Scales 299
36.C Scaled Classes and Uniformization 302
36.D The Novikov Kondo Uniformization Theorem 304
36.E Regularity Properties of Uniformizing Functions 307
36.F Uniformizing Co Analytic Sets with Large Sections 309
36.G Examples of Co Analytic Scales 310
Chapter V
Projective Sets 313
37. The Projective Hierarchy 313
37.A Basic Facts 313
37.B Examples 316
38. Projective Determinacy 322
38.A The Second Level of the Projective Hierarchy 322
38.B Projective Determinacy 325
38.C Regularity Properties 326
39. The Periodicity Theorems 327
39.A Periodicity in the Projective Hierarchy 327
39.B The First Periodicity Theorem 327
39. C The Second Periodicity Theorem 336
39.D The Third Periodicity Theorem 342
40. Epilogue 346
40.A Extensions of the Projective Hierarchy 346
40.B Effective Descriptive Set Theory 346
40.C Large Cardinals 346
40.D Connections to Other Areas of Mathematics 347
Appendix A. Ordinals and Cardinals 349
Appendix B. Well founded Relations 351
Appendix C. On Logical Notation 353
Notes and Hints 357
References 369
Symbols and Abbreviations 381
Index 387
|
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| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3/22 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV009900596 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T09:44:08Z |
| institution | BVB |
| isbn | 3540943749 0387943749 |
| language | English |
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| physical | XVIII, 402 S. graph. Darst. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
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| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spellingShingle | Kechris, Alexander S. 1946- Classical descriptive set theory Graduate texts in mathematics Ensembles, Théorie des Ensembles, théorie des ram Verzamelingen (wiskunde) gtt Set theory Deskriptive Mengenlehre (DE-588)4149180-4 gnd |
| subject_GND | (DE-588)4149180-4 |
| title | Classical descriptive set theory |
| title_auth | Classical descriptive set theory |
| title_exact_search | Classical descriptive set theory |
| title_full | Classical descriptive set theory Alexander S. Kechris |
| title_fullStr | Classical descriptive set theory Alexander S. Kechris |
| title_full_unstemmed | Classical descriptive set theory Alexander S. Kechris |
| title_short | Classical descriptive set theory |
| title_sort | classical descriptive set theory |
| topic | Ensembles, Théorie des Ensembles, théorie des ram Verzamelingen (wiskunde) gtt Set theory Deskriptive Mengenlehre (DE-588)4149180-4 gnd |
| topic_facet | Ensembles, Théorie des Ensembles, théorie des Verzamelingen (wiskunde) Set theory Deskriptive Mengenlehre |
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Teilbibliothek Mathematik & Informatik
| Signatur: |
0102 MAT 040f 2001 A 20393 Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |