TOPOLOGY WITHOUT TEARS1 SIDNEY A. MORRIS2 Version of October 14, 20073 1 c Copyright 1985-2007. No part of this book may be reproduced by any process without prior written permission from the author. If you would like a printable version of this book please e-mail your name, address, and commitment to respect the copyright (by not providing the password, hard copy or soft copy to anyone else) to [email protected] 2A version of this book translated into Persian is expected to be available soon. 3This book is being progressively updated and expanded; it is anticipated that there will be about fifteen chapters in all. If you discover any errors or you have suggested improvements please e-mail: [email protected] Contents 0 Introduction 5 0.1 Acknowledgment . 7 0.2 Readers – Locations and Professions . 7 0.3 Readers’ Compliments . 8 1 Topological Spaces 13 1.1 Topology . 14 1.2 Open Sets . 21 1.3 Finite-Closed Topology . 26 1.4 Postscript . 33 2 The Euclidean Topology 35 2.1 Euclidean Topology . 36 2.2 Basis for a Topology . 41 2.3 Basis for a Given Topology . 48 2.4 Postscript . 55 3 Limit Points 56 3.1 Limit Points and Closure . 57 3.2 Neighbourhoods . 62 3.3 Connectedness . 66 3.4 Postscript . 69 4 Homeomorphisms 70 4.1 Subspaces . 70 4.2 Homeomorphisms . 75 4.3 Non-Homeomorphic Spaces . 81 4.4 Postscript . 89 5 Continuous Mappings 90 2 CONTENTS 3 5.1 Continuous Mappings . 90 5.2 Intermediate Value Theorem . 97 5.3 Postscript . 103 6 Metric Spaces 104 6.1 Metric Spaces . 104 6.2 Convergence of Sequences . 121 6.3 Completeness . 125 6.4 Contraction Mappings . 136 6.5 Baire Spaces . 140 6.6 Postscript . 147 7 Compactness 149 7.1 Compact Spaces . 150 7.2 The Heine-Borel Theorem . 154 7.3 Postscript . 161 8 Finite Products 162 8.1 The Product Topology . 163 8.2 Projections onto Factors of a Product . 167 8.3 Tychonoff’s Theorem for Finite Products . 172 8.4 Products and Connectedness . 175 8.5 Fundamental Theorem of Algebra . 178 8.6 Postscript . 181 9 Countable Products 182 9.1 The Cantor Set . 183 9.2 The Product Topology . 185 9.3 The Cantor Space and the Hilbert Cube . 189 9.4 Urysohn’s Theorem . 196 9.5 Peano’s Theorem . 205 9.6 Postscript . 212 10 Tychonoff’s Theorem 213 10.1 The Product Topology For All Products . 214 10.2 Zorn’s Lemma . 218 10.3 Tychonoff’s Theorem . 224 10.4 Stone-Cech˘ Compactification . 239 10.5 Postscript . 246 4 CONTENTS Appendix 1: Infinite Sets 247 Appendix 2: Topology Personalities 270 Appendix 3: Chaos Theory and Dynamical Systems 277 Appendix 4: Hausdorff Dimension 306 Appendix 5: Topological Groups 319 Bibliography 343 Index 359 Chapter 0 Introduction Topology is an important and interesting area of mathematics, the study of which will not only introduce you to new concepts and theorems but also put into context old ones like continuous functions. However, to say just this is to understate the significance of topology. It is so fundamental that its influence is evident in almost every other branch of mathematics. This makes the study of topology relevant to all who aspire to be mathematicians whether their first love is (or will be) algebra, analysis, category theory, chaos, continuum mechanics, dynamics, geometry, industrial mathematics, mathematical biology, mathematical economics, mathematical finance, mathematical modelling, mathematical physics, mathematics of communication, number theory, numerical mathematics, operations research or statistics. (The substantial bibliography at the end of this book suffices to indicate that topology does indeed have relevance to all these areas, and more.) Topological notions like compactness, connectedness and denseness are as basic to mathematicians of today as sets and functions were to those of last century. Topology has several different branches — general topology (also known as point-set topology), algebraic topology, differential topology and topological algebra — the first, general topology, being the door to the study of the others. I aim in this book to provide a thorough grounding in general topology. Anyone who conscientiously studies about the first ten chapters and solves at least half of the exercises will certainly have such a grounding. For the reader who has not previously studied an axiomatic branch of mathematics such as abstract algebra, learning to write proofs will be a hurdle. To assist you to learn how to write proofs, quite often in the early chapters, I include an aside which does not form part of the proof but outlines the thought process which led to the proof. 5 6 CHAPTER 0. INTRODUCTION Asides are indicated in the following manner: In order to arrive at the proof, I went through this thought process, which might well be called the “discovery or “experiment phase. However, the reader will learn that while discovery or experimentation is often essential, nothing can replace a formal proof. There are many exercises in this book. Only by working through a good number of exercises will you master this course. I have not provided answers to the exercises, and I have no intention of doing so. It is my opinion that there are enough worked examples and proofs within the text itself, that it is not necessary to provide answers to exercises – indeed it is probably undesirable to do so. Very often I include new concepts in the exercises; the concepts which I consider most important will generally be introduced again in the text. Harder exercises are indicated by an *. Readers of this book may wish to communicate with each other regarding difficulties, solutions to exercises, comments on thisbook, and further reading. To make this easier I have created a Facebook Group called “Topology Without Tears readers. You are most welcome to join this group by sending me ([email protected]) an email requesting that. Finally, I should mention that mathematical advances are best understood when considered in their historical context. This book currently fails to address the historical context sufficiently. For the present I have had to content myself with notes on topology personalities in Appendix 2 - these notes largely being extracted from The MacTutor History of Mathematics Archive [206]. The reader is encouraged to visit the website The MacTutor History of Mathematics Archive [206] and to read the full articles as well as articles on other key personalities. But a good understanding of history is rarely obtained by reading from just one source. In the context of history, all I will say here is that much of the topology described in this book was discovered in the first half of the twentieth century. And one could well say that the centre of gravity for this period of discovery is, or was, Poland. (Borders have moved considerably.) It would be fair to say that World War II permanently changed the centre of gravity. The reader should consult Appendix 2 to understand this remark. 0.1. ACKNOWLEDGMENT 7 0.1 Acknowledgment Portions of earlier versions of this book were used at La Trobe University, University of New England, University of Wollongong, University of Queensland, University of South Australia, City College of New York, and the University of Ballarat over the last 30 years. I wish to thank those students who criticized the earlier versions and identified errors. Special thanks go to Deborah King and Allison Plant for pointing out numerous errors and weaknesses in the presentation. Thanks also go to several other colleagues including Carolyn McPhail, Ralph Kopperman, Karl Heinrich Hofmann, Rodney Nillsen, Peter Pleasants, Geoffrey Prince, Bevan Thompson and Ewan Barker who read various versions and offered suggestions for improvements. Thanks go to Rod Nillsen whose notes on chaos were useful in preparing the relevant section of Chapter 6. Particular thanks also go to Jack Gray whose excellent University of New South Wales Lecture Notes “Set Theory and Transfinite Arithmetic", written in the 1970s, influenced our Appendix on Infinite Set Theory. In various places in this book, especially Appendix 2, there are historical notes. I acknowledge two wonderful sources Bourbaki [30] and The MacTutor History of Mathematics Archive [206]. 0.2 Readers – Locations and Professions This book has been used by actuaries, astronomers, chemists, computer scientists, econometricians, economists, aeronautical, database, electrical, mechanical, software, spatial & telecommunications engineers, finance students, applied & pure mathematicians, neurophysiologists, options traders, philosophers, physicists, psychologists, software developers,spatial information scientists, and statisticians in Algeria, Argentina, Australia, Austria, Bangladesh, Bolivia, Belarus, Belgium, Belize, Brazil, Bulgaria, Cambodia, Cameroon, Canada, Chile, China, Colombia, Costa Rica, Croatia, Cyprus, Czech Republic, Denmark, Egypt, Estonia, Ethiopia, Fiji, Finland, France, Gaza, Germany, Ghana, Greece, Guyana, Hungary, Iceland, India, Indonesia, Iran, Iraq, Israel, Italy, Jamaica, Japan, Kenya, Korea, Kuwait, Lithuania, Luxembourg, Malaysia, Malta, Mauritius, Mexico, New Zealand, Nicaragua, Nigeria, Norway, Pakistan, Paraguay, Peru, Philippines. Poland, Portugal, Qatar, Romania, Russia, Serbia and Montenegro, Sierra Leone, Singapore, Slovenia, South Africa, Spain, Sri Lanka, Sudan, Sweden, Switzerland, Taiwan, Thailand, The Netherlands, The Phillipines, Trinidad and Tobago, Tunisia, Turkey, United Kingdom, Ukraine, United Arab Emirates, United States of America, Uruguay, Uzbekistan, Venezuela, and Vietnam. The book is referenced, in particular, on http://www.econphd.net/notes.htm a website 8 CHAPTER 0. INTRODUCTION designed to make known useful references for “graduate-level course notes in all core disciplines suitable for Economics students and on Topology Atlas a resource on Topology http://at.yorku.ca/topology/educ.htm. 0.3 Readers’ Compliments T. Lessley, USA: “delightful work, beautifully written; E. Ferrer, Australia: “your notes are fantastic; E. Yuan, Germany: “it is really a fantastic book for beginners in Topology; S. Kumar, India: “very much impressed with the easy treatment of the subject, which can be easily followed by nonmathematicians; Pawin Siriprapanukul, Thailand: “I am preparing myself for a Ph.D.