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  A Polish space (group) is a separable, completely metrizable topological space (group). This book is about actions of Polish groups, in connection with--or from the point of view of--the subject of descriptive set theory. Descriptive set theory is the study of definable sets and functions in Polish spaces. The basic classes of definable sets are the classes of Borel, analytic and coanalytic sets, and these constitute the main topic of the book, but the authors also consider other classes of definable sets. This will be a valuable book for all researchers in set theory and related areas.
Research monograph on set theory by two of the world's leading researchers.| PREFACE |
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vii | (1) |
| INTRODUCTION |
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viii | |
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0. DESCRIPTIVE SET THEORY |
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1 | (2) |
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3 | (10) |
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3 | (1) |
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3 | (4) |
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7 | (1) |
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1.4 Universal Polish groups |
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7 | (1) |
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1.5 Some facts about the symmetric group S(XXX) |
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8 | (3) |
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1.6 Standard Borel and Polishable groups |
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11 | (2) |
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2. ACTIONS OF POLISH GROUPS |
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13 | (20) |
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13 | (1) |
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14 | (3) |
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17 | (3) |
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20 | (2) |
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22 | (1) |
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22 | (6) |
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2.7 Universal actions for closed subgroups of S(XXX) |
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28 | (5) |
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33 | (11) |
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33 | (1) |
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3.2 Equivalence relations induced by actions |
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34 | (1) |
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34 | (1) |
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3.4 The Glimm-Effros Dichotomy |
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35 | (6) |
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3.5 Universal equivalence relations |
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41 | (3) |
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4. INVARIANT MEASURES AND PARADOXICAL DECOMPOSITIONS |
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44 | (9) |
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44 | (1) |
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4.2 Countable decompositions |
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44 | (1) |
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45 | (1) |
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46 | (1) |
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4.5 Sketch of proof of Nadkarni's Theorem |
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47 | (5) |
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4.6 Concluding remarks and problems |
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52 | (1) |
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53 | (29) |
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5.1 Finer topologies and Borel sets |
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53 | (5) |
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5.2 Topological realization of Borel G-spaces |
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58 | (9) |
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5.3 Topological realization of definable G-spaces |
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67 | (4) |
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5.4 Finer topologies on G-spaces |
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71 | (11) |
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6. MODEL THEORY AND THE VAUGHT CONJECTURE |
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82 | (16) |
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6.1 Background on the Vaught Conjecture |
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82 | (5) |
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6.2 The Topological Vaught Conjecture |
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87 | (9) |
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96 | (2) |
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7. ACTIONS WITH BOREL ORBIT EQUIVALENCE RELATIONS |
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98 | (18) |
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98 | (5) |
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7.2 Some effective considerations |
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103 | (1) |
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104 | (4) |
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108 | (2) |
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110 | (6) |
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116 | (6) |
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116 | (3) |
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8.2 Orbit cardinality for specific groups |
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119 | (3) |
| References |
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122 | (10) |
| Index |
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132 | |
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