Invariant descriptive set theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton [u.a.]
CRC Press
2009
|
| Schriftenreihe: | Pure and applied mathematics
293 |
| Schlagworte: | |
| Online-Zugang: | Klappentext Inhaltsverzeichnis |
| Beschreibung: | "A Chapman & Hall book." Includes bibliographical references (p. 361-372) and index |
| Beschreibung: | XIV, 383 S. |
| ISBN: | 9781584887935 1584887931 |
Internformat
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| 245 | 1 | 0 | |a Invariant descriptive set theory |c Su Gao |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 2009 | |
| 300 | |a XIV, 383 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Pure and applied mathematics |v 293 | |
| 500 | |a "A Chapman & Hall book." | ||
| 500 | |a Includes bibliographical references (p. 361-372) and index | ||
| 650 | 4 | |a Descriptive set theory | |
| 650 | 4 | |a Invariant sets | |
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Datensatz im Suchindex
| _version_ | 1804140695549640704 |
|---|---|
| adam_text | Exploring an active area of mathematics that studies the complexity of equivalence
relations and classification problems, Invariant Descriptive Set Theory presents
an introduction to the basic concepts, methods, and results of this theory. It brings
together techniques from various areas of mathematics, such as algebra, topology,
and logic, which have diverse applications to other fields.
After reviewing classical and effective descriptive set theory, the text studies Polish
groups and their actions. It then covers
Borei reducibility
results on
Borei,
orbit,
and general definable equivalence relations. The author also provides proofs for
numerous fundamental results, such as the Glimm-Effros dichotomy, the Burgess
trichotomy theorem, and the Hjorth turbulence theorem. The next part describes
connections with the countable model theory of
infinitan/
logic, along with Scott
analysis and the isomorphism relation on natural classes of countable models,
such as graphs, trees, and groups. The book concludes with applications to
classification problems and many benchmark equivalence relations.
By illustrating the relevance of invariant descriptive set theory to other fields of
mathematics, this self-contained book encourages readers to further explore this
very active area of research.
Features
•
Reviews classical descriptive set theory
•
Covers all aspects of Polish group actions and equivalence relations
•
Explores diverse applications in mathematics, including applications to
classification problems
•
Includes a large number of exercises at the end of most sections
•
Contains an appendix with proofs of useful results about the Gandy-
Harrington topology
Contents
Preface
xiii
I Polish Group Actions
1
1
Preliminaries
3
1.1
Polish spaces
........................... 4
1.2
The universal Urysohn space
.................. 8
1.3
Borei
sets and
Borei
functions
................. 13
1.4
Standard
Borei
spaces
...................... 18
1.5
The effective hierarchy
..................... 23
1.6
Analytic sets and
Σ}
sets
.................... 29
1.7
Coanalytic sets and
П{
sets
................... 33
1.8
The Gandy Harrington topology
................ 36
2
Polish Groups
39
2.1
Metrics on topological groups
.................. 40
2.2
Polish groups
........................... 44
2.3
Continuity of homomorphisms
................. 51
2.4
The permutation group SOo
................... 54
2.5
Universal Polish groups
..................... 59
2.6
The Graev metric groups
.................... 62
3
Polish Group Actions
71
3.1
Polish G-spaces
......................... 71
3.2
The Vaught transforms
..................... 75
3.3
Borei
G-spaces
.......................... 81
3.4
Orbit equivalence relations
................... 85
3.5
Extensions of Polish group actions
............... 89
3.6
The logic actions
......................... 92
4
Finer Polish Topologies
97
4.1
Strong Choquet spaces
..................... 97
4.2
Change of topology
....................... 102
4.3
Finer topologies on Polish G-spaces
.............. 105
4.4
Topological realization of
Borei
G-spaces
........... 109
ix
χ
II Theory of
Equivalence
Relations
115
5
Borei Reducibility
117
5.1
Borei
reductions
......................... 117
5.2
Faithful
Borei
reductions
.................... 121
5.3
Perfect set theorems for equivalence relations
......... 124
5.4
Smooth equivalence relations
.................. 128
6
The Glimm-Effros Dichotomy
133
6.1
The equivalence relation Eq
................... 133
6.2
Orbit equivalence relations embedding
£Ό
........... 137
6.3
The Harrington Kechris-Louveau theorem
.......... 141
6.4
Consequences of the Glimm-Effros dichotomy
........ 147
6.5
Actions of
cli
Polish groups
................... 151
7
Countable
Borei
Equivalence Relations
157
7.1
Generalities of countable
Borei
equivalence relations
..... 157
7.2
Hyperfinite equivalence relations
................ 160
7.3
Universal countable
Borei
equivalence relations
........ 165
7.4
Amenable groups and amenable equivalence relations
.... 168
7.5
Actions of locally compact Polish groups
........... 174
8
Borei
Equivalence Relations
179
8.1
Hypersmooth equivalence relations
............... 179
8.2
Borei
orbit equivalence relations
................ 184
8.3
A jump operator for
Borei
equivalence relations
....... 187
8.4
Examples of Fa equivalence relations
............. 193
8.5
Examples of
11°
equivalence relations
............. 196
9
Analytic Equivalence Relations
201
9.1
The Burgess trichotomy theorem
................ 201
9.2
Definable reductions among analytic equivalence relations
. . 206
9.3
Actions of standard
Borei
groups
................ 210
9.4
Wild Polish groups
....................... 213
9.5
The topological Vaught conjecture
............... 219
10
Turbulent Actions of Polish Groups
223
10.1
Homomorphisms and generic ergodicity
............ 223
10.2
Local orbits of Polish group actions
.............. 227
10.3
Turbulent and generically turbulent actions
.......... 230
10.4
The Hjorth turbulence theorem
................ 235
10.5
Examples of turbulence
..................... 239
10.6
Orbit equivalence relations and E
............... 241
Xl
III Countable
Model
Theory
245
11
Polish Topologies of Infinitary Logic
247
11.1
A review of first-order logic
................... 247
11.2
Model theory of infinitary logic
................. 252
11.3
Invariant
Borei
classes of countable models
.......... 256
11.4
Polish topologies generated by countable fragments
..... 262
11.5
Atomic models and Gs orbits
.................. 266
12
The Scott Analysis
273
12.1
Elements of the Scott analysis
................. 273
12.2
Borei
approximations of isomorphism relations
........ 279
12.3
The Scott rank and computable ordinals
........... 283
12.4
A topological variation of the Scott analysis
......... 286
12.5
Sharp analysis of SOo-orbits
................... 292
13
Natural Classes of Countable Models
299
13.1
Countable graphs
........................ 299
13.2
Countable trees
......................... 304
13.3
Countable linear
orderings
................... 310
13.4
Countable groups
........................ 314
IV Applications to Classification Problems
321
14
Classification by Example: Polish Metric Spaces
323
14.1
Standard
Borei
structures on hyperspaces
........... 323
14.2
Classification versus nonclassification
............. 329
14.3
Measurement of complexity
................... 334
14.4
Classification notions
...................... 339
15
Summary of Benchmark Equivalence Relations
345
15.1
Classification problems up to essential countability
...... 345
15.2
A roadmap of
Borei
equivalence relations
........... 348
15.3
Orbit equivalence relations
................... 350
15.4
General
Σ}
equivalence relations
................ 352
15.5
Beyond analyticity
........................ 353
A Proofs about the Gandy-Harrington Topology
355
A.I The Gandy basis theorem
.................... 355
A.2 The Gandy-Harrington topology on Xiow
........... 358
References
361
Index
373
|
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| ctrlnum | (OCoLC)150375706 (DE-599)GBV572340001 |
| dewey-full | 511.3/22 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3/22 |
| dewey-search | 511.3/22 |
| dewey-sort | 3511.3 222 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T22:04:04Z |
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| isbn | 9781584887935 1584887931 |
| language | English |
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| physical | XIV, 383 S. |
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| series2 | Pure and applied mathematics |
| spelling | Gao, Su Verfasser aut Invariant descriptive set theory Su Gao Boca Raton [u.a.] CRC Press 2009 XIV, 383 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 293 "A Chapman & Hall book." Includes bibliographical references (p. 361-372) and index Descriptive set theory Invariant sets Deskriptive Mengenlehre (DE-588)4149180-4 gnd rswk-swf Deskriptive Mengenlehre (DE-588)4149180-4 s DE-604 Pure and applied mathematics 293 (DE-604)BV000001885 293 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627466&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627466&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gao, Su Invariant descriptive set theory Pure and applied mathematics Descriptive set theory Invariant sets Deskriptive Mengenlehre (DE-588)4149180-4 gnd |
| subject_GND | (DE-588)4149180-4 |
| title | Invariant descriptive set theory |
| title_auth | Invariant descriptive set theory |
| title_exact_search | Invariant descriptive set theory |
| title_full | Invariant descriptive set theory Su Gao |
| title_fullStr | Invariant descriptive set theory Su Gao |
| title_full_unstemmed | Invariant descriptive set theory Su Gao |
| title_short | Invariant descriptive set theory |
| title_sort | invariant descriptive set theory |
| topic | Descriptive set theory Invariant sets Deskriptive Mengenlehre (DE-588)4149180-4 gnd |
| topic_facet | Descriptive set theory Invariant sets Deskriptive Mengenlehre |
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