Transcription of TOPOLOGY WITHOUT TEARS1
1 TOPOLOGY WITHOUT TEARS1 SIDNEY A. MORRISV ersion of January 2, 20112 Translations of portions of the October 2007 version of this book intoArabic (by Ms Alia Mari Al Nuaimat),Chinese (by Dr Fusheng Bai),Persian (by Dr Asef Nazari Ganjehlou),Russian (by Dr Eldar Hajilarov), andSpanish (by Dr Guillermo Pineda-Villavicencio)are now Copyright part of this book may be reproduced by any process WITHOUT priorwritten permission from the you would like a (free) printable version of this book please your name,(ii) your address [not your email address], and(iii) state explicitly your agreement to respect my copyright by not providing the password, hardcopy or soft copy of the book to anyone else. [Teachers are most welcome to use this materialin their classes and tell their students about this book but may not provide their students acopy of the book or the password.]
2 They should advise their students to email me individually.]2 This book is being progressively updated and expanded; it is anticipated that there will be aboutfifteen chapters in all. If you discover any errors or you have suggested improvements please .. Locations and Professions .. Compliments .. Author .. 151 Topological .. Sets .. TOPOLOGY .. 372 The Euclidean TOPOLOGY .. for a TOPOLOGY .. for a Given TOPOLOGY .. 583 Limit Points and Closure .. Spaces .. 912 CONTENTS35 Continuous Mappings .. Value Theorem .. 1066 Metric Spaces .. of Sequences .. Mappings .. Spaces .. Spaces .. Heine-Borel Theorem .. 1678 Finite Product TOPOLOGY .. onto Factors of a Product .. s Theorem for Finite Products .. and Connectedness .. Theorem of Algebra .. 1869 Countable Cantor Set .. Product TOPOLOGY .
3 Cantor Space and the Hilbert Cube .. s Theorem .. s Theorem .. 21910 Tychonoff s The Product TOPOLOGY For All Products .. Zorn s Lemma .. Tychonoff s Theorem .. Stone- Cech Compactification .. Postscript .. 254 Appendix 1: Infinite Sets255 Appendix 2: TOPOLOGY Personalities278 Appendix 3: Chaos Theory and Dynamical Systems286 Appendix 4: Hausdorff Dimension318 Appendix 5: Topological Groups330 Bibliography366 Index384 Chapter 0 IntroductionTopology is an important and interesting area of mathematics, the study of whichwill not only introduce you to new concepts and theorems but also put into contextold ones like continuous , to say just this is to understatethe significance of TOPOLOGY . It is so fundamental that its influence is evident inalmost every other branch of makes the study of topologyrelevant to all who aspire to be mathematicians whether their first love is (orwill be) algebra, analysis, category theory, chaos, continuum mechanics, dynamics,geometry, industrial mathematics, mathematical biology, mathematical economics,mathematical finance, mathematical modelling, mathematical physics, mathematicsof communication, number theory, numerical mathematics, operations research orstatistics.
4 (The substantial bibliography at the end of this book suffices to indicatethat TOPOLOGY does indeed have relevance to all these areas, and more.) Topologicalnotions like compactness, connectedness and denseness are as basic to mathematiciansof today as sets and functions were to those of last has several different branches general TOPOLOGY (also known as point-set TOPOLOGY ), algebraic TOPOLOGY , differential TOPOLOGY and topological algebra thefirst, general TOPOLOGY , being the door to the study of the others. I aim in this bookto provide a thorough grounding in general TOPOLOGY . Anyone who conscientiouslystudies about the first ten chapters and solves at least half of the exercises willcertainly have such a the reader who has not previously studied an axiomatic branch of mathematicssuch as abstract algebra, learning to write proofs will be a hurdle. To assist you tolearn how to write proofs, quite often in the early chapters, I include an aside whichdoes not form part of the proof but outlines the thought process which led to 0.
5 INTRODUCTIONA sides are indicated in the following manner:In order to arrive at the proof, I went through this thought process, whichmight well be called the discovery or experiment phase .However, the reader will learn that while discovery or experimentation isoften essential, nothing can replace a formal book is typset using the beautiful typesetting package, TEX, designed byDonald Knuth. While this is a very clever software package, it is my strong viewthat, wherever possible, the statement of a result and its entire proof should appearon the same page this makes it easier for the reader to keep in mind what facts areknown, what you are trying to prove, and what has been proved up to this point in aproof. So I do not hesitate to leave a blank half-page (or use subtleTEXtypesettingtricks) if the result will be that the statement of a result and its proof will then beon the one are many exercises in this book.
6 Only by working through a good numberof 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 workedexamples and proofs within the text itself, that it is not necessary to provide answersto exercises indeed it is probably undesirable to do so. Very often I include newconcepts in the exercises; the concepts which I consider most important will generallybe introduced again in the exercises are indicated by an *.Readers of this book may wish to communicate with each other regardingdifficulties, solutions to exercises, comments on this book, and further make this easier I have created a Facebook Group called TOPOLOGY WithoutTears Readers . You are most welcome to join this Group. First, search for theGroup, and then from there ask to join the , I should mention that mathematical advances are best understood whenconsidered in their historical book currently fails to address thehistorical context sufficiently.
7 For the present I have had to content myself with noteson TOPOLOGY personalities in Appendix 2 - these notes largely being extracted fromThe MacTutor History of Mathematics Archive [214]. The reader is encouraged tovisit the website The MacTutor History of Mathematics Archive [214] and to read thefull articles as well as articles on other key personalities. But a good ACKNOWLEDGMENTS7of history is rarely obtained by reading from just one the context of history, all I will say here is that much of the TOPOLOGY describedin this book was discovered in the first half of the twentieth century. And one couldwell 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 IIpermanently changed the centre of gravity. The reader should consult Appendix 2to understand this of earlier versions of this book were used at La Trobe University, Universityof New England, University of Wollongong, University of Queensland, University ofSouth Australia, City College of New York, and the University of Ballarat over thelast 30 years.
8 I wish to thank those students who criticized the earlier versionsand identified errors. Special thanks go to Deborah King and Allison Plant forpointing out numerous errors and weaknesses in the presentation. Thanks also go toseveral others, some of them colleagues, including Ewan Barker, Eldar Hajilarov, KarlHeinrich Hofmann, Ralph Kopperman, Ray-Shang Lo, , Rodney Nillsen, GuillermoPineda-Villavicencio, Peter Pleasants, Geoffrey Prince, Carolyn McPhail Sandison,and Bevan Thompson who read various versions and offered suggestions for go to Rod Nillsen whose notes on chaos were useful in preparing the relevantappendix. Particular thanks also go to Jack Gray whose excellent University of NewSouth Wales Lecture Notes Set Theory and Transfinite Arithmetic , written in the1970s, influenced our Appendix on Infinite Set various places in this book, especially Appendix 2, there are historical acknowledge two wonderful sources Bourbaki [32] and The MacTutor History ofMathematics Archive [214].
9 Initially the book was typset using Donald Knuth s beautiful and powerful the book was expanded and colour introduced, this was translated into 5 is based on my 1977 book Pontryagin duality and the structure oflocally compact abelian groups Morris [172]. I am grateful to Dr Carolyn McPhailSandison for typesetting this book in TEXfor me, a decade 0. Locations and ProfessionsThis book has been, or is being, used by professors, graduate students, undergraduatestudents, high school students, and retirees, some of whom are studying to be, areor were, accountants, actuaries, applied and pure mathematicians, astronomers,chemists, computer graphics, computer scientists, econometricians, economists,aeronautical, database, electrical, mechanical, software, space, spatial and telecommunicationsengineers, finance experts, neurophysiologists, nutritionists, options traders, philosophers,physicists, psychiatrists, psychoanalysts, psychologists, sculptors, software developers,spatial information scientists, and statisticians in Algeria, Argentina, Australia, Austria,Bangladesh, Bolivia, Belarus, Belgium, Belize, Brazil, Bulgaria, Cambodia, Cameroon,Canada, Chile, Gabon, People s Republic of China, Colombia, Costa Rica, Croatia,Cyprus.
10 Czech Republic, Denmark, Egypt, Estonia, Ethiopia, Fiji, Finland, France,Gaza, Germany, Ghana, Greece, Greenland, Guatemala, Guyana, Hungary, Iceland,India, Indonesia, Iran, Iraq, Israel, Italy, Jamaica, Japan, Kenya, Korea, Kuwait,Liberia, Lithuania, Luxembourg, Malaysia, Malta, Mauritius, Mexico, New Zealand,Nicaragua, Nigeria, Norway, Pakistan, Panama, Paraguay, Peru, Poland, Portugal,Qatar, Romania, Russia, Serbia, Sierra Leone, Singapore, Slovenia, South Africa,Spain, Sri Lanka, Sudan, Sweden, Switzerland, Taiwan, Thailand, The Netherlands,The Phillippines, Trinidad and Tobago, Tunisia, Turkey, United Kingdom, Ukraine,United Arab Emirates, United States of America, Uruguay, Uzbekistan, Venezuela,and book is referenced, in particular, on awebsite designed to make known useful references for graduate-level course notesin all core disciplines suitable for Economics students and on TOPOLOGY Atlas aresource on ComplimentsT.
