A course in point set topology PDF
Preview A course in point set topology
Undergraduate Texts in Mathematics John B. Conway A Course in Point Set Topology Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege,Williamstown,MA,USA AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada RuthCharney,BrandeisUniversity,Waltham,MA,USA IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA RogerE.Howe,YaleUniversity,NewHaven,CT,USA DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA JillPipher,BrownUniversity,Providence,RI,USA FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA AmieWilkinson,UniversityofChicago,Chicago,IL,USA Undergraduate Texts in Mathematics are generally aimed at third- and fourth- year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches. The books include motivation that guides the reader to an apprecia- tionofinterrelationsamongdifferentaspectsofthesubject. Theyfeatureexamples thatillustratekeyconceptsaswellasexercisesthatstrengthenunderstanding. Forfurthervolumes: http://www.springer.com/series/666 John B. Conway A Course in Point Set Topology 123 JohnB.Conway DepartmentofMathematics TheGeorgeWashingtonUniversity Washington,DC,USA ISSN0172-6056 ISSN2197-5604(electronic) ISBN978-3-319-02367-0 ISBN978-3-319-02368-7(eBook) DOI10.1007/978-3-319-02368-7 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013951651 MathematicsSubjectClassification(2010):54A05,54E35,54D20,58C07 ©SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright. AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered andexecuted onacomputer system, forexclusive usebythepurchaser ofthework. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter. Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade. Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) For Ann How could I be so lucky! Preface Pointset topology was my first love in mathematics. I took the courseas an undergraduateatLoyolaUniversityinNew Orleansandmyprofessor,Harry Fledderman, told me to go to the library and solve all the problems in the book while he tutored the other student who had signed up for the course. (Yes, Iknowitsoundsstrangetoday,buttherewereonlytwostudentsinthe course.) I kept a notebook with my solutions, and once a week I reported for his inspection of my work. I felt like a real mathematician learning real mathematics. Ithad a greatinfluence onme andmade me realizehow much Iwantedtobeamathematician. EvennowIcan’ttellyouwhethertheloveI havefor point set topologywas the cause of this feeling or whether that love wasa consequenceof this learningstyle. I was disappointedto later discover that researchin this area had mostly petered out. I found equally attractive research areas in which to sow my oats, but I always retained this youthful love affair. You can probably guess that I have long wanted to write a book on this topic, but other things took precedence. I am glad that was the case because now I think I have a better approach. I had an epiphany about halfwaythroughmycareerwhenIrealizedIdidn’thavetoteachmystudents everythingIhadlearnedaboutthesubjectathand. Ilearnedmathematicsin schoolthatIneverusedagain,andnotjustbecausethosethingswereinareas in which I never did research. At least part of this, I suspect, was because some of my teachers hadn’t had this insight. Another reason is that many authors write textbooks as though they are writing a monograph directed at other faculty rather than thinking of the students as the audience. Also, mathematics refines and refreshes itself with time. Certain topics that were important at the inception of an area fade in significance, and some that are useful in various areas today must be added. Other topics are important, but only if you are one of that small percentage who specialize in a specific part of research; such things should not be taught to everyone who takes an vii viii Preface introductory course. In addition, when a subject is developing, there is an emphasis on finding the intellectual boundaries of the concepts. Unless that viewpoint is abandoned when the subject is taught, it results in a greater prominence of pathology. An examination of early texts in any subject will reveal such an emphasis. With time, however, it is crucial to decide what should be taught in an introductory course such as the kind this book is written for. I see the purpose of a course in point set topology as giving the student a set of tools. The material is used in almost every part of mathematics. In addition, I have come to believe that in teaching, it is best to go fromthe particulartothe moregeneral,anapproachthathasdominatedmy presentations in the last several texts I have written. To begin with, that is the way mathematics evolves. In addition, my experience is that all but the very best students find such an approach more digestible. The present book reflects my belief in this approach. I see this text as aimed at an undergraduate audience that has had cal- culus and been exposed to the ideas of basic set theory like subsets, unions, intersections,functions,andlittleelse. Nevertheless,Ithinkitadvisablethat they have had at least a semester of analysis and been properly exposed to convergence and other topological notions in the real line. (I have included some appendices to help bridge the gap, but I am sure this will not suffice forall.) Ialsothink pointsettopologyis anexcellentplacetobeginlearning how to digest and write proofs. Thus, I tend to go slowly at the start of the book, including more detail than is needed for a seasoned student and even more than I include later in the text. Following my philosophy of beginning with the particular, I start with metric spaces. I believe that these are far easier to connect with students’ experience. Theyalsoseemtometobethemoreprevalenttopologicalspaces usedinotherareasandarethereforeworthextraemphasis. Chapter2defines and develops abstract topological spaces, with metric spaces as the source of inspiration. I narrow the discussion by quickly restricting the focus to Hausdorff spaces. Needless to say, some of the more elementary arguments in topological spaces are the same as those in metric spaces. There is no problemhere; I just referstudents to the metric space proofandinvite them tocarryouttheanalogousargument,whichinmostcasesisalmostidentical. Chapter 3 concentrates on continuous real-valued functions. My belief is that the continuous functions on a space are more important than the underlying space. Maybe that’s because I’m an analyst. I know that much of moderntopology concentratesonthe underlying geometryof a space, but surely that must be saved until after the student has encountered the need. Biographies. In this book I continue the practice, startedin a previous work, of including short biographicalnotes when a mathematician’s result is mentioned. There is no scholarship on my part in this as all the material Preface ix is from secondary sources, principally what I could find on the Web. In particular, I made heavy use of the site at the University of St Andrews http://www-history.mcs.st-andrews.ac.uk/history/BiogIndex.html and Wikipedia. I continue my practice of emphasizing personal aspects of these lives over their mathematical achievements, especially if there is some- thing there that interests me. Style. The longerI ama mathematicianandthe moreI write, the more I ask myself questions about style. There is a difference between what you write and how you speak. That’s true in mathematics just as it is outside ofmathematics. I think I write more informally thanmany mathematicians, but there are some rules I try to follow even though they are not universal. OnesuchruleisthatwithessentiallynoexceptionsIusesymbolslike , ,>, ∈ ⊆ and so on as verbs and only as verbs. (For example, translates as “is an ∈ element of” and not as “in.”) I think this consistency expedites reading. I experimented a long time ago with using such symbols only as prepositions butquicklydecidedthiswasawkward. WhenIamhavingadiscussioninmy office, I will frequently use them both ways, but when I write I try to stick with using them as verbs. So don’t forget to read them that way. Long ago I realized that every result is not a theorem. The label “The- orem” is reservedfor the truly important results. It’s not that those labeled “Proposition” are unimportant, but they may be more routine or, perhaps, they just don’t have the impact on the development of the subject at hand. A corollary is a direct consequence of a proposition or a theorem. A lemma is a result whose usefulness is usually limited to the proof of the next result. A Word to Students. If you want to learn mathematics, you cannot approachit as a spectator sport; sitting on the edge of the pool and dipping yourtoeinwillnotgetyouintothesubject. Youmustjumpintoitwithboth feet, commityourself,anddoalotofdirtyworkandsplashingaroundbefore you can enter the profession at any level. In the course of this book there aremany places where I leaveproofs to readersas exercises;do them. When I give such an exercise, I think it is well within the scope of your ability provided you understood the concepts. Doing those exercises will confirm that you understand what has come before; if you cannot do them, it may mean you overlookedsomething and should go back. Throughout the text you will see words like Verify! and Why? I am trying to put a speed bump in your reading. I want you to be sure you understand what was just said. Sometimes my exercises ask a question. A basic part of mathematics is deciding whether something is true andthen provingit. Mathematicians are constantly trying to discover whether something is true and are seldom, if ever, presented with a known truth and asked to prove it. So when you see such an exercise and you think something is true, you must prove it; if you think it false, you must find a counterexample. (The ability to manufacture examples is a precious talent that you should cultivate.)
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