Table Of Content

M o n o g r a f i e M a t e m a t y c z n e Instytut Matematyczny Polskiej Akademii Nauk (IMPAN) Volume 71 (New Series) Founded in 1932 by S. Banach, B. Knaster, K. Kuratowski, S. Mazurkiewicz, W. Sierpinski, H. Steinhaus Managing Editor: Przemysław Wojtaszczyk, IMPAN and Warsaw University Editorial Board: Jean Bourgain (IAS, Princeton, USA) Tadeusz Iwaniec (Syracuse University, USA) Tom Körner (Cambridge, UK) Krystyna Kuperberg (Auburn University, USA) Tomasz Łuczak (Poznán University, Poland) Ludomir Newelski (Wrocław University, Poland) Gilles Pisier (Université Paris 6, France) Piotr Pragacz (Institute of Mathematics, Polish Academy of Sciences) Grzegorz Świątek (Pennsylvania State University, USA) Jerzy Zabczyk (Institute of Mathematics, Polish Academy of Sciences) Volumes 31–62 of the series Monografie Matematyczne were published by PWN – Polish Scientific Publishers, Warsaw Lev Bukovský The Structure of the Real Line Lev Bukovský Institute of Mathematics University of P.J. Šafárik Jesenná 5 040 01 Košice Slovakia [email protected] 2010 Mathematics Subject Classification: 03E15, 03E17, 03E25, 03E35, 03E50, 03E60, 03E65, 26A21, 28A05, 28A99, 54D99, 54G15, 54H05 ISBN 978-3-0348-0005-1 e-ISBN 978-3-0348-0006-8 DOI 10.1007/978-3-0348-0006-8 Library of Congress Control Number: 2011923518 © Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com To Zuzana Contents 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 viii Contents 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 Σ1 Sets . . . . . . . . . . . . . . . . . . . . . . . 238 2 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 – Bartoszyn´skiTheorem . . . . . . . . . . . . 271 7.5 Cichon´ 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 Contents ix 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