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Universitext Editorial Board (North America): S. Axler K.A. Ribet Volker Runde A Taste of Topology With 17 Figures VolkerRunde DepartmentofMathematical andStatisticalSciences UniversityofAlberta Edmonton,Alberta CanadaT6G2G1 [email protected] EditorialBoard (NorthAmerica): K.A.Ribet S.Axler MathematicsDepartment MathematicsDepartment UniversityofCaliforniaatBerkeley SanFranciscoStateUniversity Berkeley,CA94720-3840 SanFrancisco,CA94132 USA USA [email protected] [email protected] MathematicsSubjectClassification(2000):54-01,55-01 LibraryofCongressControlNumber:2005924410 ISBN-10:0-387-25790-X Printedonacid-freepaper. ISBN-13:978-0387-25790-7 ©2005SpringerScience+BusinessMedia,Inc. All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermissionofthepublisher(SpringerScience+BusinessMedia,Inc.,233SpringStreet,New York,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis. Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornot theyaresubjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. (MVY) 9 8 7 6 5 4 3 2 1 springeronline.com Volker Runde A Taste of Topology March 14, 2005 Springer Berlin Heidelberg NewYork HongKong London Milan Paris Tokyo Preface Ifmathematicsisalanguage,thentakingatopologycourseattheundergradu- ateleveliscrammingvocabularyandmemorizingirregularverbs:anecessary, but not always exciting exercise one has to go through before one can read greatworksofliteratureintheoriginallanguage,whosebeautyeventually—in retrospect—compensates for all the drudgery. Set-theoretic topology leaves its mark on mathematics not so much through powerful theorems (even though there are some), but rather by pro- viding a unified framework for many phenomena in a wide range of mathematical disciplines. An introductory course in topology is necessarily concept heavy; the nature of the subject demands it. If the instructor wants to flesh out the concepts with examples, one problem arises immediately in an un- dergraduate course: the students don’t yet have a mathematical background broadenoughthatwouldenablethemtounderstand“natural”examples,such as those from analysis or geometry.Most examples in such a course therefore tend to be of the concocted kind: constructions, sometimes rather intricate, that serve no purpose other than to show that property XY is stronger than propertyYX whereasthe converseis false.Thereis the veryrealdangerthat studentscomeoutofatopologycoursebelievingthatfreelyjugglingwithdefi- nitionsandcontrivedexamplesiswhatmathematics—oratleasttopology—is all about. The present book grew out of lecture notes for Math 447 (Elementary Topology)at the University of Alberta, a fourth-yearundergraduate course I taughtinthewinterterm2004.I hadoriginallyplannedtouse[Simmons 63] as a text, mainly because it was the book from which I learned the material. Since there were some topics I wanted to cover, but that were not treated in [Simmons 63], I started typing my own notes and making them available on the Web, and in the end I wound up writing my own book. My audience included second-year undergraduates as well as graduate students, so their mathematical background was inevitably very varied. This fact has greatly influencedtheexposition,inparticulartheselectionofexamples.Ihavemade an effort to present examples that are, firstly, not self-serving and, secondly, vi Preface accessible for students who have a background in calculus and elementary algebra, but not necessarily in real or complex analysis. It is clear that an introductory topology text only allows for a limited degree of novelty.Most topics coveredin this book can be found in any other book on the subject. I have thus tried my best to make the presentation as fresh and accessible as possible, but whether I have succeeded depends very much on my readers’ tastes. Besides, in a few points, this books treats its material differently than—to my knowledge, at least—any other text on the subject. • Baire’s theorem is derived from Bourbaki’s Mittag-Leffler theorem; • Nets are extensively used, and, in particular, we give a fairly intuitive proof—using nets—of Tychonoff’s theorem due to Paul R. Chernoff [Chernoff 92]; • The complex Stone–Weierstraß theorem is obtained via Silvio Machado’s short and elegant approach[Machado 77]. With a given syllabus and a limited amount of classroom time, every instructor in every course has to make choices on what to cover and what to omit. These choices will invariably reflect his or her own tastes and biases,in particular, when it comes to omissions. The topics most ostensibly omitted fromthis book are:filters and uniform spaces.Isimply find nets,with allthe parallels between them and sequences, far more intuitive than filters when it comes to discussing convergence (others may disagree). Treating uniform spacesinanintroductorycourseisaproblem,inmyopinion,duetothelackof elementary, yet natural,examples that aren’t metric spaces in the first place. Any book, even if there is only one author named on the cover, is to some extent an accomplishment of several people. This one is no exception, and I would like to thank Eva Maria Krause for her thorough and insightful proofreadingofthe entiremanuscript.Of course,without my students—their feedbackandenthusiasm—thisbookwouldnothavebeenwritten.Ihopethat takingthecoursewasasmuchfunforthemasteachingitwasforme,andthat they had A Taste of Topology that will make their appetite for mathematics grow in the years to come. Volker Runde Edmonton, March 14, 2005 Contents Preface ........................................................ v List of Symbols ................................................ ix Introduction................................................... 1 1 Set Theory ................................................ 5 1.1 Sets and Functions ...................................... 5 1.2 Cardinals............................................... 13 1.3 Cartesian Products ...................................... 17 Remarks.................................................... 20 2 Metric Spaces ............................................. 23 2.1 Definitions and Examples................................. 23 2.2 Open and Closed Sets.................................... 28 2.3 Convergence and Continuity .............................. 34 2.4 Completeness ........................................... 40 2.5 Compactness for Metric Spaces............................ 52 Remarks.................................................... 59 3 Set-Theoretic Topology.................................... 61 3.1 Topological Spaces—Definitions and Examples .............. 61 3.2 Continuity and Convergence of Nets ....................... 72 3.3 Compactness............................................ 79 3.4 Connectedness .......................................... 89 3.5 Separation Properties ....................................100 Remarks....................................................107 4 Systems of Continuous Functions ..........................109 4.1 Urysohn’s Lemma and Applications........................109 4.2 The Stone–Cˇech Compactification .........................116 viii Contents 4.3 The Stone–Weierstraß Theorems ..........................121 Remarks....................................................129 5 Basic Algebraic Topology..................................133 5.1 Homotopy and the Fundamental Group ....................133 5.2 Covering Spaces.........................................148 Remarks....................................................154 A The Classical Mittag-Leffler Theorem Derived from Bourbaki’s.................................................157 B Failure of the Heine–Borel Theorem in Infinite- Dimensional Spaces........................................161 C The Arzelà–Ascoli Theorem ...............................165 References.....................................................169 Index..........................................................171 List of Symbols (0), 68 Bx, 65 (cid:1)·(cid:1), 24 (cid:1)·(cid:1)1, 24 c, 16 T(cid:1)·(cid:1)∞, 24 C, 5 S{S :S ∈S},8 C∞, 86 {S :S ∈S},8 C([0,1]), 24 ∈, 5 C(X,Y), 42 ∞, 34 Cb(X,Y), 42 ∈/, 5 C0(X,F),126 ∂QS, 33 cl, 67 Q{S :S ∈S}, 18 i∈ISi, 18 d, 24 ∼, 134 diam, 44 (cid:5), 136 dim, 40 ⊂, 6 dist, 34 (cid:1), 6 distF, 122 ∅, 5 2κ, 16 φα, 144 ℵ0,16 f|A, 10 f(A), 10 (a,b),6 f−1(B), 10 [a,b], 6 f ◦g,11 (a,b], 6 F, 24 [a,b), 6 A∩B, 8 f−1, 12 A∪B, 8 f∗, 142 A\B, 8 f :S →T, 10 F(S,Y), 65 Ar,R[x0], 135 βX, 118 [γ], 141 Bn, 143 γ1(cid:12)γ2, 98 Br(x0), 28 γ−1, 98 Br[x0], 30 B(S,Y), 24 H(Ω),159 x List of Symbols idS,10 |S|≤|T|, 14 |S|<|T|, 14 limαxα, 74 Spec(R),63 limn→∞xn, 35 Sn−1,90 L(U), 58 S2, 9 S×T, 9 µ, 95 SI, 18 Sn, 17 N, 5 N0, 5 T, 61 Nx, 29 TC, 65 Nf,C,(cid:3), 65 T∞, 86 Nx, 64 V(I),63 π, 8 π1(X,x0), 138 χn, 95 πn(X,x0),155 (xα)α, 74 p, 63 (xα)α∈A, 74 P(S), 7 xα→x, 74 P(X,x0), 138 (X,d), 24 P(X;x0,x1), 138 X∞, 86 (xn)∞n=1, 10 Q, 5 (xn)∞n=m, 10 xn→x,35 R, 5 x(cid:15)y,18 R(f;P,ξ), 74 (“X“,T),”62 ” X˜,T˜ ,p , 149 S, 30 (x,y),9 |S|=|T|, 13 |S|≥|T|, 14 |S|>|T|, 14 Yx, 94 ◦ S, 34 Z, 5