The Structure of the Real Line
The rapid development of set theory in the last fifty years, mainly in obtaining plenty of independence results, strongly influenced an understanding of the structure of the real line. This book is devoted to the study of the real line and its subsets tak
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M a t e ma 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 . . . . . . . . . . . . . . . . . . . . . . Historical and Bibliographical Notes . . . . . . . . . . . . . . . . . . . .
21 36
2 The Real Line 2.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.2
Topology of the Real Line . . . . . . . . . . . . . . . . . . . . . .
46
2.3 2.4
Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . Expressing a Real by Natural Numbers . . . . . . . . . . . . . . .
55 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 3.4
Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Convergence of Functions . . . . . . . . . . . . . . . . . . . . . . 105
3.5
Baire Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Historical and Biblio
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