Preface ........................................................ xi
1 Introduction
1.1 Set Theory .............................................. 2
1.2 Topological Preliminaries .............................. 21
Historical and Bibliographical Notes ........................ 36
2 The Real Line
2.1 The Definition ......................................... 39
2.2 Topology of the Real Line .............................. 46
2.3 Existence and Uniqueness ............................... 55
2.4 Expressing a Real by Natural Numbers ................... 62
Historical and Bibliographical Notes ........................ 70
3 Metric Spaces and Real Functions
3.1 Metric and Euclidean Spaces ............................ 74
3.2 Polish Spaces .......................................... 86
3.3 Borel Sets ............................................. 97
3.4 Convergence of Functions .............................. 105
3.5 Baire Hierarchy ....................................... 116
Historical and Bibliographical Notes ....................... 124
4 Measure Theory
4.1 Measure ............................................... 127
4.2 Lebesgue Measure ...................................... 139
4.3 Elementary Integration ................................ 145
4.4 Product of Measures, Ergodic Theorem .................. 154
Historical and Bibliographical Notes ....................... 159
5 Useful Tools and Technologies
5.1 Souslin Schemes and Sieves ............................ 162
5.2 Pointclasses .......................................... 171
5.3 Boolean Algebras ...................................... 182
5.4 Infinite Combinatorics ................................ 194
5.5 Games Played by Infinitely Patient Players ............ 207
Historical and Bibliographical Notes ....................... 213
6 Descriptive Set Theory
6.1 Borel Hierarchy ....................................... 216
6.2 Analytic Sets ......................................... 222
6.3 Projective Hierarchy .................................. 230
6.4 Co-analytic and Σ12 Sets .............................. 238
Historical and Bibliographical Notes ....................... 246
7 Decline and Fall of the Duality
7.1 Duality of Measure and Category ....................... 250
7.2 Duality Continued ..................................... 258
7.3 Similar not Dual ...................................... 266
7.4 The Fall of Duality Bartoszyński Theorem .............. 271
7.5 Cichoń Diagram ........................................ 282
Historical and Bibliographical Notes ....................... 290
8 Special Sets of Reals
8.1 Small Sets ............................................ 293
8.2 Sets with Nice Subsets ................................ 307
8.3 Sequence Convergence Properties ....................... 317
8.4 Covering Properties ................................... 332
8.5 Coverings versus Sequences ............................ 342
8.6 Thin Sets of Trigonometric Series ..................... 353
Historical and Bibliographical Notes ....................... 368
9 Additional Axioms
9.1 Continuum Hypothesis and Martin's Axiom ............... 375
9.2 Equalities, Inequalities and All That ................. 383
9.3 Assuming Regularity of Sets of Reals .................. 396
9.4 The Axiom of Determinacy .............................. 405
Historical and Bibliographical Notes ....................... 413
10 Undecidable Statements
10.1 Projective Sets ....................................... 415
10.2 Measure Problem ....................................... 422
10.3 The Linear Ordering of the Real Line .................. 430
10.4 Reversing the Order of Integration .................... 440
10.5 Permitted Sets of Trigonometric Series ................ 446
Historical and Bibliographical Notes ....................... 453
11 Appendix
11.1 Sets, Posets, and Trees ............................... 455
11.2 Rings and Fields ...................................... 465
11.3 Topology and the Real Line ............................ 473
11.4 Some Logic ............................................ 475
11.5 The Metamathematics of the Set Theory ................. 483
Bibliography .................................................. 493
Index of Notation ............................................. 521
Index ......................................................... 525
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