Topology

315 Pages·2015·4.139 MB·English

by  Marco Manetti (auth.)

Marco Manetti
Topology
T
T
X
X
E
E
TT
II
NN
123
UU
UNITEXT - La Matematica per il 3+2
Volume 91
Editor-in-chief
A. Quarteroni
Series editors
L. Ambrosio
P. Biscari
C. Ciliberto
M. Ledoux
W.J. Runggaldier
More information about this series at http://www.springer.com/series/5418
Marco Manetti
Topology
123
MarcoManetti
Department ofMathematics
Sapienza -UniversitàdiRoma
Rome
Italy
TranslatedbySimonG.Chiossi,UFBA—UniversidadeFederaldaBahia,Salvador(Brazil).
TranslationfromtheItalianlanguageedition:Topologia,MarcoManetti,©Springer-Verlag
Italia, Milano 2014.All rights reserved.
ISSN 2038-5722 ISSN 2038-5757 (electronic)
UNITEXT- La Matematica peril3+2
ISBN978-3-319-16957-6 ISBN978-3-319-16958-3 (eBook)
DOI 10.1007/978-3-319-16958-3
LibraryofCongressControlNumber:2014945348
SpringerChamHeidelbergNewYorkDordrechtLondon
©SpringerInternationalPublishingSwitzerland2015
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
foranyerrorsoromissionsthatmayhavebeenmade.
Coverdesign:SimonaColombo,GiochidiGrafica,Milano,Italy
Printedonacid-freepaper
SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia
(www.springer.com)
Preface
To the Student
Thistextbookoffersaprimeringeneraltopology(point-settopology),togetherwith
an introduction to algebraic topology. It is meant primarily for students with a
mathematical background that is usually taught in the first year of undergraduate
degrees in Mathematics and Physics.
Point-settopologyisthelanguageinwhichaconsiderablepartofmathematicsis
written.Itisnotanaccidentthattheoriginalname‘analytictopology’wasreplaced
by‘generaltopology’,amoreapttermforthatpartoftopologythatisusedbythevast
majorityofmathematiciansandisfundamentalinmanyareasofmathematics.Over
timeitsunabatedemploymenthashadaconstantpolishingeffectonitstheoremsand
definitions,thusrenderingitanextraordinarilyelegantsubject.Thereisnodoubtthat
point-settopologyhasasignificantformativevalue,inthatitforcesthebrain—and
trains it at the same time—to handle extremely abstract objects, defined solely by
axioms. In studying on this book, you will experience hands-on that the point-set
topology resembles a language more than a theory. There are endless terms and
definitionstobelearnt,amyriadoftheoremswhoseproofisoftenrathereasy,only
occasionallyexceeding20lines.Thereare,obviously,alsodeepandfar-from-trivial
results,suchasthetheoremsofBaire,AlexanderandTychonov.
Thepartonalgebraictopology,detailsofwhichwewillgiveinChap.9together
withthemandatorymotivations,isdevotedtothestudyofhomotopy,fundamental
groups and covering spaces.
Iincludedaround500exercisesinthetext:tryingtosolvethemwithdedication
is the best way to attain a firm hold on the matter, adapt it to your own way of
thinkingandalsolearntodeveloporiginalideas.Someexercisesaresolveddirectly
inthetext,eitherinfulloralmost.Theyarecalled‘Examples’,andtheirimportance
should not be underestimated: understanding them is the correct way to
make abstract notions concrete. Exercises marked with ~, instead, are solved in
Chap. 16.
v
vi Preface
It is a matter of fact that the best way to learn a new subject is by attending
lectures, or studying on books, and trying to understand definitions, theorems and
the interrelationships properly. At the same time you should solve the exercises,
withoutthefearofmakingmistakes,andthencomparethesolutionswith theones
in the text, or those provided by teacher, classmates or the internet.
This book also proposes a number of exercises marked with , which I per-
sonallybelievetobeharderthanthetypicalexamquestion.Theseexercisesshould
thereforebetakenasendeavourstointelligence,andincentivestobecreative:they
requirethatweabandonourselvestonewsynergiesofideasandaccepttobeguided
by subtler analogies, rather than trail patiently along a path paved by routine ideas
and standard suggestions.
To the Lecturer
In the academic years 2004–2005 and 2005–2006, I taught a lecture course called
‘Topology’ for the Bachelor’s degree in Mathematics at University of Rome
‘La Sapienza’. The aim was to fit the newly introduced programme specifications
for mathematical teaching in that part of the syllabus traditionally covered in
‘Geometry 2’ course of the earlier 4-year degrees. The themes were carefully
chosensotokeepintoaccountononesidetheformativeandculturalfeaturesofthe
singletopics,ontheothertheirusefulnessinthestudyofmathematicsandresearch
alike. Some choices certainly break with a long-standing and established tradition
of topology teaching in Italy, and with hindsight I suspect they might have been
elicited by my own research work in algebra and algebraic geometry. I decided it
would be best to get straight to the point and state key results and definitions as
early as possible, thus fending off the terato(po)logical aspects.
From the initial project to the final layout of my notes, I tried to tackle the
conceptual obstacles gradually, and make both theory and exercises as interesting
andentertainingaspossibleforstudents.WhetherIachievedthesegoalsthereader
will tell.
Thebackgroundnecessarytobenefitfromthebookisstandard,astaughtinfirst-
yearMathsandPhysicsundergraduatecourses.Solidknowledgeofthelanguageof
sets,oflinearalgebra,basicgrouptheory,thepropertiesofrealfunctions,seriesand
sequences from ‘Calculus’ are needed. The second chapter is dedicated to the
arithmeticofcardinalnumbersandZorn’slemma,twopivotalprerequisitesthatare
not always addressed during the first year: it will be up to the lecturer to decide—
after assessing the students’ proficiency—whether to discuss these topics or not.
The material present here is more than sufficient for 90 hours of lectures and
exercise classes, even if, nowadays, mathematics syllabi tend to allocate far less
time totopology.Inorder tohelp teachersdecide whattoskip Iindicated with the
symbolyancillarytopics,whichmaybeleftoutatfirstreading.Ithastobesaid,
though,thatChaps.3–6(withtheexceptionofthesectionsdisplayingy),formthe
backbone of point-set topology and, as such, should not be excluded.
Preface vii
The bibliography is clearly incomplete and lists manuals that I found most
useful,plusaselectionofresearcharticlesandbookswherethewillingstudentcan
find further information about the topics treated, or mentioned in passing, in this
volume.
Acknowledgments
I am grateful to Ciro Ciliberto and Domenico Fiorenza for reading earlier versions
andforthetipstheygaveme.IwouldliketothankFrancescaBonadeiofSpringer-
VerlagItaliaforhelpingwiththefinallayout,andallthestudents(‘victims’)ofmy
lectures on topology for the almost-always-useful and relevant observations that
allowed me to amend and improve the book.
This volume is based on the second Italian edition. Simon G. Chiossi has done
an excellent work of translation; he also pointed out a few inaccuracies and pro-
posed minor improvements. To him and the staff at Springer, I express here my
heartfelt gratitude.
Future updates can be found at
http://www.mat.uniroma1.it/people/manetti/librotopology.html.
Rome, February 2015 Marco Manetti
Contents
1 Geometrical Introduction to Topology. . . . . . . . . . . . . . . . . . . . . 1
1.1 A Bicycle Ride Through the Streets of Rome. . . . . . . . . . . . 1
1.2 Topological Sewing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Notion of Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Homeomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Facts Without Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Notations and Basic Concepts. . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Induction and Completeness. . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Cardinality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 The Axiom of Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Zorn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6 The Cardinality of the Product . . . . . . . . . . . . . . . . . . . . . . 36
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Topological Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Topological Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Interior of a Set, Closure and Neighbourhoods . . . . . . . . . . . 43
3.3 Continuous Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Subspaces and Immersions. . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Topological Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7 Hausdorff Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Connectedness and Compactness. . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Connected Components. . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Covers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
ix
x Contents
4.4 Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 Wallace’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6 Topological Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.7 Exhaustions by Compact Sets. . . . . . . . . . . . . . . . . . . . . . . 83
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 Topological Quotients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1 Identifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Quotient Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Quotients by Groups of Homeomorphisms. . . . . . . . . . . . . . 92
5.4 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.5 Locally Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.6 The Fundamental Theorem of Algebra y . . . . . . . . . . . . . . 101
6 Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1 Countability Axioms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4 Compact Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.5 Baire’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.6 Completions y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.7 Function Spaces and Ascoli-Arzelà Theorem y . . . . . . . . . . 122
6.8 Directed Sets and Nets (Generalised Sequences) y. . . . . . . . 125
7 Manifolds, Infinite Products and Paracompactness. . . . . . . . . . . . 129
7.1 Sub-bases and Alexander’s Theorem. . . . . . . . . . . . . . . . . . 129
7.2 Infinite Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.3 Refinements and Paracompactness y . . . . . . . . . . . . . . . . . 133
7.4 Topological Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.5 Normal Spaces y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.6 Separation Axioms y. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8 More Topics in General Topology y. . . . . . . . . . . . . . . . . . . . . . 143
8.1 Russell’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.2 The Axiom of Choice Implies Zorn’s Lemma. . . . . . . . . . . . 144
8.3 Zermelo’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.4 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.5 The Compact-Open Topology. . . . . . . . . . . . . . . . . . . . . . . 151
8.6 Noetherian Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.7 A Long Exercise: Tietze’s Extension Theorem . . . . . . . . . . . 157
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159