A Taste of Topology

182 Pages·2005·2.346 MB·English

by  Volker Runde (auth.), S Axler, K.A. Ribet (eds.)

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 mathe-
matical 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.Ihadoriginallyplannedtouse[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 intu-
itive 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 in-
structor 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 Arzel`a–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