A First Course
Textbook in Problems
O. Y. Viro, O. A. Ivanov,
introdu es algebrai topology via the fundamental group and
ov-eringspa es,and providesaba kgroundontopologi aland smooth
manifolds. Itiswrittenmainlyforstudentswithalimitedexperien e
in mathemati s, but determined to study thesubje ta tively. The
material is presented in a on ise form, proofs are omitted.
Theo-rems, however,are formulatedin detail, and thereaderis expe ted
Genre, Contents and Style of the Book
The oreofthebookisthematerialusuallyin ludedintheTopologypart
ofthetwoyearGeometryle ture ourseattheMathemati alDepartment
of St. Petersburg University. It was omposed by Vladimir Abramovi h
Rokhlin inthe sixties and has almost not hanged sin e then.
We believe this isthe minimum topologythat must bemastered by any
student who has de ided to be ome a mathemati ian. Students with
resear h interests in topology and related elds will surely need to go
beyond this book, but it may serve as a starting point. The book
in- ludes basi materialongeneral topology, introdu es algebrai topology
via its most lassi al and elementary part, the theory of the
fundamen-talgroupand overingspa es, andprovidesaba kgroundontopologi al
and smooth manifolds. It is written mainlyfor students with a limited
experien einmathemati s,butwho aredeterminedtostudythe subje t
a tively.
The orematerialispresentedina on iseform;proofsareomitted.
The-orems, however, are formulated indetail. Wepresent them as problems
and expe t the reader totreat them asproblems. Mostof the theorems
areeasyto ndelsewherewith ompleteproofs. Webelievethataserious
attempttoproveatheoremmust bethe rst rea tiontoitsformulation.
It shouldpre ede lookingfor a book where the theorem is proved.
On the other hand, we want to emphasize the role of formulations. In
the early stages of studying mathemati s it is espe ially important to
takeea h formulationseriously. Weintentionallyfor ea reader tothink
about ea h simple statement. We hope that this will make the book
in onvenient for mere skimming.
The ore material is enhan ed by many problems of various sorts and
additional pie es of theory. Although they are losely related to the
mainmaterial,they anbe(andusuallyare)keptoutsideofthestandard
le ture ourse. Theseenhan ements anbere ognizedbywidermargins,
Theproblems,whi hdonot ompriseseparatetopi sandareintended
ex lu-sivelytobeexer ises,aretypesetwithsmallfa e.Someofthemareveryeasy
andin ludedjusttoprovideadditionalexamples. FewproblemsarediÆ ult.
Theyaretoindi aterelationswithotherpartsofmathemati s,showpossible
dire tionsofdevelopmentofthesubje t,orjust satisfyanambitiousreader.
Problems, whose solutions seemto be themost diÆ ult(from the authors'
viewpoint),aremarkedwithastar, asinmanyotherbooks.
Further, we want to deliveradditional pie es of theory (with respe t to
the orematerial)to more motivated and advan edstudents. Maybe,a
mathemati ian, whodoesnot work inthe eldsgeometri in avor, an
a ord the luxury not to know some of these things. Maybe, students
studying topology an postpone this material to their graduate study.
We would like to in lude this in graduate le ture ourses. However,
quite often it does not happen, be ause most of the topi s of this sort
are rather isolated from the ontents of traditional graduate ourses.
They are important, but more related to the material of the very rst
topology ourse. In the book these topi s areintertwined withthe ore
material and exer ises, but are distinguishable: they are typeset, like
these lines, with large fa e, large margins, theorems and problems in
them arenumerated inaspe ialmannerdes ribedbelow.
Exer isesandillustrativeproblemstotheadditionaltopi saretypeset
withevenwidermarginsandmarkedinadi erentway.
Thus, the whole book ontains four layers:
the ore material,
exer ises and illustrative problems tothe ore material,
additionaltopi s,
exer ises and illustrative problems toadditionaltopi s.
The text of the ore material is typeset with large fa e and smallest
margins.
Thetext ofproblems elaboratingonthe orematerial istypeset withsmall
fa e andlargermargins.
The text of additional topi s is typeset is typeset with large fa e and
slightlysmallermarginsastheproblemselaboratingonthe orematerial.
Thetextofproblemsillustratingadditionaltopi sistypesetwithsmall
fa eandthelargestmargins.
Therefore the book looks like a Russian folklore doll, matreshka
om-posed of several dolls sitting inside ea h other. We apologize for being
The whole text of the book is divided into se tions. Ea h se tion is
divided into subse tions. Subse tions are not numerated. Ea h of them
is devoted to a single topi and onsists of de nitions, ommentaries,
theorems, exer ises,problems, and riddles.
By a riddle we mean a problem of a spe ial sort: its solution is not
ontained in the formulation. One has to guess a solution, rather than
dedu e it.
0.A. Theorems, exer ises, problems and riddles belonging to the ore
material are marked with pairs onsisting of the number of se tion and
a letter separated with a dot. The letter identi es the item inside the
se tion.
0.1. Exer ises,problems,andriddles,whi harenotin ludedinthe ore,but
are loselyrelatedtoit(andtypeset withsmallfa e) aremarkedwith pairs
onsistingofthenumberofthese tionandthenumberoftheiteminsidethe
se tion. Thenumbersinthepairareseparatedalsobyadot.
Theorems, exer ises, problems and riddles related to additional topi s
are enumerated independentlyinside ea hse tionand denoted similarly.
0:A. The only di eren e is that the omponents of pairs marking the
items areseparated bya olon (rather thandot).
Weassumethatthereaderisfamiliarwithnaivesettheory,butanti ipate
that this familiarity may be super ial. Therefore at points where set
theory is espe ially ru ialwe makeset-theoreti digressions maintained
in the same style as the rest of the book.
Advi e to the Reader
Sin e the book ontains a summary of elementary topology, you may
use thebookwhile preparing foranexamination(espe ially,if the exam
redu es to solving a olle tion of problems). However, if you attend
le tures on the subje t, it would be mu h wiser to read the book prior
to the le tures and prove theorems beforethe le turer gives the proofs.
We think that a reader who is able to prove statements of the ore of
the book, does not need to solve all the other problems. It would be
reasonable insteadto look through formulations and on entrate on the
mostdiÆ ultproblems. ThemorediÆ ultthe theoremsofthe maintext
seemtoyou,themore arefullyyoushould onsiderillustrativeproblems,
and the less time youshould waste with problems marked with stars.
Keep in mind that sometimes a problem whi h seems to be diÆ ult is
problem whi h suggests a return to the theorem, on e you are armed
with the lemmas.
Mostofourillustrativeproblemsareeasytoinvent,and,moreover,ifyou
study the subje t seriously, it is always worthwhile to invent problems
of this sort. To develop this style ofstudying mathemati swhile solving
our problemsoneshouldattempttoinventone'sown problemsandsolve
them (it does not matter if they are similar to ours or not). Of ourse,
Foreword iii
Genre, Contentsand Styleof the Book iii
Advi e tothe Reader v
Part 1. General Topology 1
Chapter 1. Generalities 3
1. Topology in a Set 3
De nition of Topologi al Spa e 3
Simplest Examples 3
The Most ImportantExample: RealLine 4
Using New Words: Points, Open and Closed Sets 4
Set-Theoreti Digression. De Morgan Formulas 5
Being Open or Closed 5
CantorSet 6
Chara terization of Topology inTerms of Closed Sets 6
Topologyand Arithmeti Progressions 7
Neighborhoods 7
2. Bases 7
De nition of Base 7
Bases for Plane 8
When aColle tion of Sets is a Base 8
Subbases 8
In nity of the Set of Prime Numbers 9
Hierar hy of Topologies 9
3. Metri Spa es 9
De nition and First Examples 9
FurtherExamples 10
Balls and Spheres 11
Subspa es of a Metri Spa e 11
Surprising Balls 11
Segments (What Is Between) 12
Bounded Sets and Balls 12
Norms and Normed Spa es 12
Metri Topology 13
Metrizable Topologi alSpa es 13
Equivalent Metri s 13
Operations with Metri s 14
Distan e Between Pointand Set 15
Distan e Between Sets 15
4. Subspa es 16
Relativity of Openness 16
Agreement on Notationsof Topologi al Spa es 17
5. Position of a Point with Respe t to a Set 17
Interior, Exterior and Boundary Points 17
Interior and Exterior 18
Closure 18
Frontier 19
Closure and Interiorwith Respe t toa FinerTopology 19
Properties of Interior and Closure 19
Chara terization of Topology by Closure orInterior Operations 20
Dense Sets 21
Nowhere DenseSets 21
LimitPointsand IsolatedPoints 22
Lo allyClosed Sets 22
6. Set-Theoreti Digression. Maps 22
Maps and the Main Classesof Maps 22
Image and Preimage 23
Identity and In lusion 24
Composition 24
Inverse and Invertible 25
Submappings 25
7. Continuous Maps 25
De nition and Main Properties of Continuous Maps 25
Reformulations of De nition 26
More Examples 26
Behaviorof Dense Sets 27
Lo alContinuity 27
Properties of Continuous Fun tions 28
Spe ial About Metri Case 28
Fun tions onCantor Set and Square-FillingCurves 29
Sets De ned by Systems of Equations and Inequalities 30
Set-Theoreti Digression. Covers 31
FundamentalCovers 31
8. Homeomorphisms 32
De nition and Main Properties of Homeomorphisms 32
Homeomorphi Spa es 32
Role of Homeomorphisms 32
More Examples of Homeomorphisms 33
Examples of Homeomorphi Spa es 34
Examples of Nonhomeomorphi Spa es 37
Information (Without Proof) 38
Embeddings 38
Information 39
Chapter 2. Topologi al Properties 40
9. Conne tedness 40
De nitions of Conne tedness and First Examples 40
Conne ted Sets 40
Properties of Conne ted Sets 41
Conne ted Components 41
TotallyDis onne ted Spa es 42
Frontier and Conne tedness 42
BehaviorUnder Continuous Maps 42
Conne tedness onLine 43
Intermediate Value Theorem and Its Genralizations 44
Dividing Pan akes 44
Indu tion onConne tedness 44
Appli ations to HomeomorphismProblem 45
10. Path-Conne tedness 46
Paths 46
Path-Conne ted Spa es 46
Path-Conne ted Sets 47
Path-Conne ted Components 47
Path-Conne tedness Versus Conne tedness 48
Polygon-Conne tedness 49
11. Separation Axioms 49
Hausdor Axiom 50
Limitsof Sequen e 50
Coin iden e Set and Fixed PointSet 50
Hereditary Properties 51
The First Separation Axiom 51
The Third Separation Axiom 52
The FourthSeparation Axiom 52
Niemytski'sSpa e 53
Urysohn Lemmaand TietzeTheorem 53
12. Countability Axioms 54
Set-Theoreti Digression. Countability 54
Se ond Countability and Separability 55
Embeddingand Metrization Theorems 56
Bases ata Point 56
First Countability 56
SequentialApproa hto Topology 57
SequentialContinuity 57
13. Compa tness 58
Compa tness inTerms of ClosedSets 59
Compa t Sets 59
Compa t Sets Versus Closed Sets 59
Compa tness and Separation Axioms 60
Compa tness inEu lidean Spa e 60
Compa tness and Maps 61
Norms inR n
62
Closed Maps 62
14. Lo al Compa tness and Para ompa tness 62
Lo alCompa tness 62
One-Point Compa ti ation 63
Proper Maps 64
Lo allyFinite Colle tionsof Subsets 64
Para ompa tSpa es 65
Para ompa tness and Separation Axioms 65
Partitionsof Unity 65
Appli ation: Making Embeddings fromPie es 66
15. Sequential Compa tness 66
SequentialCompa tness Versus Compa tness 66
In Metri Spa e 66
Completeness and Compa tness 67
Non-Compa t Balls inIn nite Dimension 67
p-Adi Numbers 68
Indu tion onCompa tness 68
Spa es of Convex Figures 69
Problems for Tests 69
Chapter 3. Topologi al Constru tions 72
16. Multipli ation 72
Set-Theoreti Digression. Produ t of Sets 72
Produ t ofTopologies 73
Topologi alProperties of Proje tions and Fibers 73
Cartesian Produ ts of Maps 74
Properties of Diagonaland Graph 74
Topologi alProperties of Produ ts 75
Representation of Spe ial Spa es asProdu ts 75
17. Quotient Spa es 76
Set-Theoreti Digression. Partitionsand Equivalen e Relations 76
QuotientTopology 77
Topologi alProperties of QuotientSpa es 78
Set-Theoreti Digression. Quotients and Maps 78
Continuity of Quotient Maps 79
Closed Partitions 79
Open Partitions 79
Toolsfor Des ribing Partitions 80
Entran eto the Zoo 81
Transitivity of Fa torization 83 MobiusStrip 83 Contra ting Subsets 83 FurtherExamples 84 Klein Bottle 84 Proje tive Plane 85
YouMay Have Been Provoked to Perform an IllegalOperation 85
Set-Theoreti Digression. Sums of Sets 85
Sums of Spa es 85
Atta hing Spa e 86
Basi Surfa es 87
19. Proje tive Spa es 88
Real Proje tive Spa e of Dimension n 88
Complex Proje tive Spa eof Dimension n 89
Quaternion Proje tive Spa esand Cayley Plane 89
20. Topologi al Groups 89
Algebrai Digression. Groups 89
Topologi alGroups 90
Self-Homeomorphisms Makinga Topologi al GroupHomogeneous 91
Neighborhoods 92 Separaion Axioms 92 CountabilityAxioms 93 Subgroups 93 Normal Subgroups 94 Homomorphisms 95 Lo alIsomorphisms 95 Dire t Produ ts 96
21. A tions of Topologi al Groups 97
A tions of Group in Set 97
Continuous A tions 97
Orbit Spa es 97
Homogeneous Spa es 98
22. Spa es of Continuous Maps 98
Sets of Continuous Mappings 98
Topologi alStru tures onSet of Continuous Mappings 98
Topologi alProperties of Spa esof Continuous Mappings 99
Metri Case 99
Intera tions WithOther Constru tions 100
Mappings XY !Z and X !C(Y;Z) 101
Part 2. Algebrai Topology 102
Continuous Deformations of Maps 104
Homotopy asMap and Familyof Maps 104
Homotopy asRelation 105
Straight-LineHomotopy 105
Two Natural Properties of Homotopies 106
Stationary Homotopy 106
Homotopies and Paths 107
Homotopy of Paths 107
24. Homotopy Properties of Path Multipli ation 108
Multipli ation of Homotopy Classes of Paths 108
Asso iativity 108
Unit 109
Inverse 109
25. Fundamental Group 110
De nition of FundamentalGroup 110
Why Index 1? 110
High Homotopy Groups 111
Cir ular loops 111
The Very First Cal ulations 112
FundamentalGroup of Produ t 113
Simply-Conne tedness 113
FundamentalGroup of a Topologi al Group 114
26. The Role of Base Point 114
Overview of the Roleof Base Point 114
De nition of TranslationMaps 115
Properties of T
s
115
Role of Path 115
High Homotopy Groups 116
In Topologi al Group 116
27. Covering Spa es 117
De nition 117
Lo alHomeomorphismsVersus Coverings 117
Numberof Sheets 118
More Examples 118
Universal Coverings 119
Theorems on Path Lifting 119
High-DimensionalHomotopy Groupsof Covering Spa e 121
28. Cal ulationsofFundamentalGroups UsingUniversal
Coverings 121
FundamentalGroup of Cir le 121
FundamentalGroup of Proje tive Spa e 122
FundamentalGroups of Bouquet of Cir les 122
Algebrai Digression. Free Groups 123
Universal Covering forBouquet of Cir les 124
Indu ed Homomorphisms 125
FundamentalTheorem of HighAlgebra 127
Generalization of Intermediate Value Theorem 127
Winding Number 128
Borsuk-Ulam Theorem 128
30. Covering Spa es via Fundamental Groups 129
Homomorphisms Indu ed by Covering Proje tions 129
Numberof Sheets 130 Hierar hy of Coverings 130 Automorphisms of Covering 131 Regular Coverings 131 Existen e of Coverings 131 Lifting Maps 131
Chapter 5. More Appli ations and Cal ulations 132
31. Retra tions and Fixed Points 132
Retra tions and Retra ts 132
FundamentalGroup and Retra tions 133
Fixed-PointProperty. 133
32. Homotopy Equivalen es 134
Homotopy Equivalen e asMap 134
Homotopy Equivalen e asRelation 135
Deformation Retra tion 135
Examples 135
Deformation Retra tion Versus Homotopy Equivalen e 136
Contra tibleSpa es 136
Fundamental Groupand Homotopy Equivalen es 137
33. Cellular Spa es 138
De nition of CellularSpa es 138
First Examples 140
More Two-DimensionalExamples 141
Topologi alProperties of CellularSpa es 142
Embeddingto Eu lidean Spa e 143
One-Dimensional CellularSpa es 143
Euler Chara teristi 144
34. Fundamental Group of a Cellular Spa e 145
One-Dimensional CellularSpa es 145
Generators 145
Relators 145
Writing Down Generators and Relators 146
Fundamental Groupsof Basi Surfa es 147
Seifert -vanKampen Theorem 148
35. One-Dimensional Homology and Cohomology 148
Des ription of H
1
(X) inTerms of Free Cir ularLoops 149
Integer Cohomology and Maps to S 1
151
One-Dimensional Homology Modulo 2 152
Part 3. Manifolds 154
Chapter 6. Bare Manifolds 156
36. Lo ally Eu lidean Spa es 156
De nition of Lo allyEu lidean Spa e 156
Dimension 157
Interior and Boundary 157
37. Manifolds 159
De nition of Manifold 159
Components of Manifold 160
Making New Manifolds out of Old Ones 160
Double 161
Collars and Bites 161
38. Isotopy 162
Isotopy of Homeomorphisms 162
Isotopy of Embeddingsand Sets 162
Isotopies and Atta hing 164
Conne ted Sums 164
39. One-Dimensional Manifolds 164
Zero-Dimensional Manifolds 164
Redu tion toConne ted Manifolds 165
Examples 165
Statements of MainTheorems 165
Lemma on1-Manifold Covered with Two Lines 166
Without Boundary 166
With Boundary 167
Consequen es of Classi ation 167
Mapping Class Groups 167
40. Two-Dimensional Manifolds 167
Examples 167
Ends and Odds 168
Closed Surfa es 169
Triangulationsof Surfa es 170
Two Properties of Triangulationsof Surfa es 170
S heme of Triangulation 171
Examples 172
Familiesof Polygons 172
Operations onFamilyof Polygons 173
Topologi aland Homotopy Classi ationof Closed Surfa es 174
Re ognizing Closed Surfa es 175
Orientations 175
Simply Conne ted Surfa es 177
41. One-Dimensional mod2-Homology of Surfa es 177
Polygonal Paths on Surfa e 177
Subdivisions of Triangulation 177
Bringing Loops toGeneral Position 179
Cutting Surfa e AlongCurve 180
Curves onSurfa es and Two-FoldCoverings 181
One-Dimensional Z 2 -Cohomology of Surfa e 181 One-Dimensional Z 2 -Homology of Surfa e 182 Poin areDuality 182
One-Sided and Two-Sided SimpleClosed Curves onSurfa es 182
Orientation Covering and First Stiefel-WhitneyClass 182
RelativeHomology 182
42. Surfa es Beyond Classi ation 182
Genus of Surfa e 183
Systems of disjoint urves ona surfa e 183
Polygonal Jordanand S hon ies Theorems 183
Polygonal Annulus Theorem 183
Dehn Twists 183
Coverings of Surfa es 183
Bran hed Coverings 183
Mapping Class Group of Torus 183
Braid Groups 183
43. Three-Dimensional Manifolds 183
Poin areConje ture 184
Lens Spa es 184
Seifert Manifolds 184
Fibrations over Cir le 184
Heegaard Splittingand Diagrams 184
Chapter 7. Smooth Manifolds 185
44. Analyti Digression:
Di erentiable Fun tions in Eu lidean Spa e 186
Di erentiabilityand Di erentials 186
Derivative AlongVe tor 187
Main Propertiesof Di erential 187
Higher OrderDerivatives 187
C r
-Maps 188
Di eomorphisms 189
Inverse Fun tionTheorem 189
Impli it Fun tion Theorem 189
C r -Fun tions 190 Useful C 1 -Fun tion 190
Appli ations of Bell-Shape Fun tion 190
C r
Motivation: Topologi al Stru ture viaContinuous Fun tions 191
Di erentialSpa es 192
Di erentialStru ture of a Metri Spa e 193
Di erentialSubspa es 194
C r
-Stru tures onSubspa e of Metri Spa e 195
Di erentiableMaps 195
Di eomorphisms 196
Di erentiale Embeddings 196
Semi ubi Parabola 197
46. Constru ting Di erential Spa es 197
Multipli ation of Di erentiable Spa es 197
Quotient Di erential Spa es 198
Classi al LieGroups and Homogeneous Spa es 199
Spa e of n-PointSubsets of Surfa e 199
Tori Varieties 199
47. Smooth Manifolds 199
C r
-Manifolds 199
Manifolds with Corners 200
TraditionalApproa h toSmooth Manifolds 200
Equivalen eof the Two Approa hes 202
Revision of Boundary 203
Revision of Multipli ation 203
Revision of Di erentiableMaps 203
Rank of Mapping 204
Di erentialTopology 204
Submanifolds 204
48. Immersions and Embeddings 205
Immersions 205
Di erentiableEmbeddings 206
Immersions Versus Embeddings 207
Embeddability toEu lidean Spa es 207
49. Tangent Ve tors 208
Coordinate De nition 209
Digression on EinsteinNotations 210
Di erentiationof Fun tions 210
Di erentialof Map 210
Tangent Bundle 210
Tangent Ve tors in Eu lidean Spa e 211
Ve tors as Velo ities 211
50. Ve tor Bundles 211
General Terminologyof Fibrations 211
Trivialand Lo allyTrivial 211
Indu ed Fibrations 211
Ve tor Bundles 211
Tautologi alBundles 211
Homotopy Classi ationof Ve tor Bundles 211
Low-Dimensional 211
51. Orientation 211
Linear Algebra Digression: Orientations of Ve torSpa e 212
Related Orientations 212
Orientation of Ve tor Bundle 212
Orientation and Orientabilityof Smooth Manifold 212
Orientation of Boundary 212
Orientation Covering 212
Proje tive Spa es 212
52. Transversality and Cobordisms 212
Sard Theorem 212
Transversality 212
Embeddingto R 2n+1
212
Normal Bundle and Tubular Neighborhood 212
Pontryagin Constru tion 212 Degree of Map 212 Linking Numbers 212 Hopf Invariant 212 Thom Constru tion 212 Cobordisms 212
portant omponents: thelanguageofset-theoreti topology,whi htreats
thebasi notionsrelatedto ontinuity. Thetermgeneraltopology means:
this isthe topology that is needed and used by most mathemati ians.
As a resear h eld, it was ompleted a long timeago. Its permanent
us-age inthe apa ity ofa ommonmathemati allanguagehas polished its
system of de nitions and theorems. Nowadays studying general
topol-ogy really resembles studying a language rather than mathemati s: one
needs to learn a lot of new words, while proofs of all theorems are
ex-tremely simple. On the other hand, the theorems are numerous. It is
not surprising: they play the role of rules regulating usageof words.
Wehavetowarn students,forwhomthis isoneofthe rstmathemati al
subje ts. Do not hurryto fall inlove with it tooseriously, donot let an
imprinting happen. This eld may seam to be harming, but it is not
very a tive. It hardly provides as mu h roomfor ex iting new resear h
Generalities
1. Topology in a Set
De nition of Topologi al Spa e
Let X be aset. Let bea olle tionof its subsetssu h that:
(a) the union ofafamilyof sets, whi h areelementsof, belongs to;
(b) the interse tion of a nite family of sets, whi h are elements of ,
belongsto ;
( ) the empty set ? and the whole X belong to.
Then
is alled atopologi al stru ture orjust a topology 1
in X;
the pair (X;) is alleda topologi al spa e;
an element of X is alled a point of this topologi al spa e;
anelementofis alledanopenset ofthetopologi alspa e(X;).
The onditionsinthede nitionaboveare alledtheaxiomsof topologi al
stru ture.
Simplest Examples
A dis rete topologi alspa e is aset with the topologi al stru ture whi h
onsists of all the subsets.
1.A. Che kthatthisisatopologi alspa e,i.e.,allaxiomsoftopologi al
stru ture hold true.
An indis rete topologi al spa e is the opposite example, in whi h the
topologi al stru ture is the most meager. It onsists onlyof X and ?.
1.B. This is a topologi alstru ture, is itnot?
Here arelesstrivialexamples.
1
Thus is important: itis alled bythesamewordasthewhole bran h of
mathe-mati s. Of ourse,thisdoesnotmeanthat oin ideswiththesubje toftopology,
1.1. Let X be the ray [0;+1), and onsists of ?, X, and all the rays
(a;+1)witha0. Provethat isatopologi alstru ture.
1.2. LetX beaplane. Let onsistof?,X,andallopendiskswith enter
attheorigin. Is thisatopologi alstru ture?
1.3. Let X onsist of four elements: X =fa;b; ;dg. Whi h of the
follow-ing olle tionsof itssubsetsare topologi alstru tures inX, i.e.,satisfythe
axiomsoftopologi alstru ture:
(a) ?,X,fag,fbg,fa; g,fa;b; g,fa;bg;
(b) ?,X,fag,fbg,fa;bg,fb;dg;
( ) ?,X,fa; ;dg,fb; ;dg?
Thespa eof 1.1is alled anarrow. Wedenotethespa e of1.3(a)by4pT.
Itisasortoftoyspa emadeof4points. Bothofthesespa es,aswellasthe
spa eof1.2,arenotimportant,butprovidegood simpleexamples.
The Most Important Example: Real Line
Let X be the set R of all real numbers, be the set of unions of all
intervals (a;b) with a;b2R.
1.C. Che k if satis esthe axioms of topologi alstru ture.
This isthetopologi alstru turewhi hisalwaysmeantwhenR is
onsid-eredasatopologi alspa e(unlessothertopologi alstru tureisexpli itly
spe i ed). This spa e is alled usually the real line and the stru ture is
referred to asthe anoni al or standard topology inR.
1.4. LetX beR, and onsistsof emptyset andallthein nitesubsetsof
R. Isatopologi alstru ture?
1.5. LetX beR, and onsistsofemptysetand omplementsofall nite
subsetsofR. Isatopologi alstru ture?
Thespa eof1.5isdenotedbyR
T
1
and alled thelinewithT
1
-topology.
1.6. Let(X;)beatopologi alspa eandY bethesetobtainedfromX by
addingasingleelementa. Is
ffag[U : U 2g[f?g
atopologi alstru tureinY?
Using New Words: Points, Open and Closed Sets
Re all that, for a topologi al spa e (X;), elements of X are alled
points,and elementsof are alled open sets. 2
2
1.D. Reformulate the axioms of topologi al stru ture using the words
open set whereverpossible.
A set F 2 X is said to be losed in the spa e (X;) if its omplement
XrF is open (i.e., XrF 2).
Set-Theoreti Digression. De Morgan Formulas
1.E. Let fA
g
2
be an arbitrary family of subsets of a set X. Prove
that Xr [ 2 A = \ 2 (X rA ) (1) Xr \ 2 A = [ 2 (XrA ): (2)
Formula (2) is dedu ed from (1) in onestep, is it not? These formulasare
nonsymmetri ases of asingle formulation,whi h ontains in asymmetri
waysetsandtheir omplements,unionsand interse tions.
1.7. Riddle. Find su haformulation.
Being Open or Closed
1.F Properties of Closed Sets. Provethat:
(a) the interse tion of any olle tionof losed sets is losed;
(b) union of any nitenumberof losed sets is losed;
( ) empty set and the wholespa e (i.e., the underlyingset of the
topo-logi al stru ture) are losed.
Noti e thatthepropertyofbeing losedisnotanegationoftheproperty
of being open.
1.G. Findexamples of sets, whi h
(a) are both open, and losed simultaneously;
(b) are neitheropen, nor losed.
1.8. Giveanexpli itdes riptionof losedsetsin
(a) adis retespa e;
(b) anindis rete spa e; ( ) thearrow; (d) 4pT; (e) R T1 .
Con epts of losed and open sets are similar in a number of ways. The
main di eren e is that the interse tion of an in nite olle tion of open
sets does not have to be ne essarily open, while the interse tion of any
olle tion of losed sets is losed. Along the same lines, the union of an
in nite olle tionof losed sets isnot ne essarily losed, whilethe union
of any olle tionof open sets is open.
1.9. Provethat thehalf-open interval[0;1)is neitheropennor losed inR,
but anbepresentedaseithertheunionof losedsetsorinterse tionofopen
sets.
1.10. Provethat everyopen set of thereal line is aunionof disjointopen
intervals.
1.11. ProvethatthesetA=f0g[ 1 n 1 n=1 is losedinR. Cantor Set
LetK bethesetofrealnumberswhi h anbepresentedassumsofseries
of the form P 1 k=1 a k 3 k with a k
=0 or2. In other words, K is the set of
real numbers whi h in the positional system with base 3 are presented
as0:a 1 a 2 :::a k ::: withoutdigit1.
1:A. Finda geometri des riptionofK.
1:A:1. Prove that
(a) K is ontained in[0;1℄,
(b) K doesnotinterse t 1 3 ; 2 3 ,
( ) K doesnotinterse t 3s+1 3 k ; 3s+2 3 k
foranyintegersk and s.
1:A:2. Present K as [0;1℄ with an in nite family of open intervals
removed.
1:A:3. Tryto drawK.
The setK is alled theCantor set. It hasalotof remarkable properties
and isinvolvedinnumerous problemsbelow.
1:B. Provethat K is a losed setinthereal line.
Chara terization of Topology in Terms of Closed Sets
1.12. Prove that if a olle tion F of subsets of X satis es the following
onditions:
(a) theinterse tionofanyfamilyofsetsfrom F belongstoF;
(b) theunionofany nitenumbersetsfromF belongstoF;
( ) ?andX belongto F,
thenF istheset ofall losed setsofatopologi alspa e(whi h one?).
Topology and Arithmeti Progressions
1.14*. ConsiderthefollowingpropertyofasubsetF ofthesetN ofnatural
numbers: there exists N 2 N su h that F does not ontain an arithmeti
progressionoflengthgreaterthanN. Prove,thatsubsetswiththisproperty
togetherwiththewholeN forma olle tionof losedsubsetsinsometopology
in N.
Solvingthis problem,youprobablyare notableto avoidthefollowing
om-binatorialtheorem.
1.15 Van der Waerden's Theorem*. Foreveryn2N thereexistsN2
N su hthatforanyAf1;2;:::;Ng,eitherAorf1;2;:::;NgrA ontains
anarithmeti progressionoflengthn.
Neighborhoods
By a neighborhood of a point one means any open set ontaining this
point. Analysts and Fren h mathemati ians (following N. Bourbaki)
prefer a wider notion of neighborhood: they use this word for any set
ontaining aneighborhood inthe sense above.
1.16. Giveanexpli itdes riptionofallneighborhoodsofapointin
(a) adis retespa e;
(b) anindis rete spa e;
( ) thearrow;
(d) 4pT.
2. Bases
De nition of Base
Usually the topologi al stru ture is presented by des ribing its part,
whi his suÆ ient tore over thewhole stru ture. A olle tionof open
sets is alled abase for a topology if ea h nonempty open set isa union
of sets of . For instan e, allintervals forma base for the real line.
2.1. Aretheredi erenttopologi alstru tureswiththesamebase?
2.2. Find somebasesoftopologyof
(a) adis retespa e;
(b) anindis retespa e;
( ) thearrow;
(d) 4pT.
Bases for Plane
2.4. Provethat anybaseofthe anoni altopologyofR anbediminished.
Considerthefollowingthree olle tionsofsubsetsofR 2
:
2
whi h onsists of all possible open disks (i.e., disks without its
boundary ir les);
1
whi h onsists of allpossibleopensquares (i.e., squares without
theirsidesandverti es)withsidesparalleltothe oordinate axis;
1
whi h onsistsofallpossibleopensquareswithsidesparalleltothe
bise torsofthe oordinateangles.
(Squares of 1
and 1
arede ned byinequalitiesmaxfjx aj;jy bjg<
andjx aj+jy bj<respe tively.)
2.5. Provethat everyelementof 2
isaunionofelementsof 1
.
2.6. Provethatinterse tionofanytwoelementsof 1
isaunionofelements
of 1
.
2.7. Provethat ea h ofthe olle tions 2
, 1
, 1
is abase for some
topo-logi al stru turein R 2
, andthat thestru tures de ned by these olle tions
oin ide.
When a Colle tion of Sets is a Base
2.A. A olle tion of open sets is a base for the topology, i for any
opensetUandanypointx2UthereisasetV 2su hthatx2V U.
2.B. A olle tionof subsets ofa set X isabase for some topologyin
X,i X is aunion ofsets of andinterse tionof any two sets of isa
union of sets in.
2.C. Show that the se ond ondition in2.B (on interse tion) is
equiva-lent to the following: the interse tion of any two sets of ontains,
to-getherwithanyofitspoints,somesetof ontainingthispoint( f.2.A).
Subbases
Let(X;)beatopologi alspa e. A olle tionofitsopensubsetsis alled
asubbase for,providedthe olle tion
=fV jV =\ k i=1 W i ;W i 2;k2Ng
ofall niteinterse tionsofsets belongingto isabase for.
2.8. Provethat foranyset X a olle tion ofitssubsetsisasubbaseofa
In nity of the Set of Prime Numbers
2.9. Provethatallin nitearithmeti progressions onsistingofnatural
num-bersform abaseforsometopologyin N.
2.10. Usingthistopologyprovethatthesetofallprimenumbersisin nite.
(Hint: otherwisetheset f1gwould beopen(?!) )
Hierar hy of Topologies
If
1
and
2
aretopologi alstru turesinasetX su hthat
1 2 then 2
is said to be ner than
1 , and 1 oarser than 2 . For instan e,
among all topologi alstru tures in the same set the indis rete topology
is the oarsest topology, and the dis rete topologyisthe nest one, is it
not?
2.11. Show that T
1
-topology(see Se tion 1) is oarser than the anoni al
topologyintherealline.
2.12. Riddle. Let
1 and
2
bebasesfortopologi alstru tures
1 and
2
in aset X. Find ne essaryand suÆ ient onditionfor
1 2 in termsof thebases 1 and 2
withoutexpli itreferringto
1 and
2
( f.2.7).
Bases de ning the same topologi alstru ture are said tobe equivalent.
2.D. Riddle. Formulate a ne essary and suÆ ient ondition for two
bases to be equivalent without expli it mentioning of topologi al
stru -tures de ned by the bases. (Cf. 2.7: bases 2 , 1 , and 1 must satisfy
the ondition you are lookingfor.)
3. Metri Spa es
De nition and First Examples
A fun tion :X X ! R
+
=fx 2 R j x 0g is alled a metri (or
distan e) inX, if
(a) (x;y)=0,i x=y;
(b) (x;y)=(y;x) forevery x;y 2X;
( ) (x;y)(x;z)+(z;y) for every x;y;z 2X.
The pair (X;), where is a metri in X, is alled ametri spa e. The
ondition ( ) is triangle inequality.
3.A. Prove that forany set X
:XX !R
+
: (x;y)7! (
is a metri . 3.B. Prove that R R !R + :(x;y)7!jx yj isa metri . 3.C. Provethat R n R n !R + :(x;y)7! p P n i=1 (x i y i ) 2 isa metri .
Metri s 3.Band 3.Care always meantwhenR and R n
are onsidered as
metri spa es unless anothermetri is spe i ed expli itly. Metri 3.Bis
a spe ial ase of metri 3.C. Thesemetri s are alled Eu lidean.
Further Examples 3.1. Provethat R n R n !R + :(x;y)7!max i=1;:::;n jx i y i jisametri . 3.2. Provethat R n R n !R + :(x;y)7! P n i=1 jx i y i jisametri . Metri s in R n
introdu ed in 3.C{3.2 are in luded in in nite series of the
metri s (p) : (x;y)7! n X i=1 jx i y i j p 1 p ; p1: 3.3. Provethat (p)
isametri foranyp1.
3.3.1 Holder Inequality. Prove that
n X i=1 x i y i n X i=1 x p i ! 1=p n X i=1 y q i ! 1=q ifx i ;y i 0,p;q>0and 1 p + 1 q =1. Metri of3.C is (2) , metri of3.2 is (1)
, andmetri of3.1 anbe denoted
by (1)
andadjoinedtotheseriessin e
lim p!+1 n X i=1 a p i 1 p =maxa i ;
foranypositivea
1 ,a 2 ,:::,a n .
3.4. Riddle. Howisthisrelatedto 2 , 1 , and 1 fromSe tion 2?
Forareal numberp1denoteby l (p)
theset ofsequen esx =fx
i g
i=1;2;:::
su hthattheseries P 1 i=1 jxj p onverges.
3.5. Provethat for any twoelements x;y 2 l (p) the series P 1 i=1 jx i y i j p
onvergesandthat
(x;y)7! 1 X i=1 jx i y i j p 1 p ; p1 (p)
Balls and Spheres
Let (X;) be a metri spa e, let a be its point, and let r be a positive
real number. The sets
D r (a)=fx2X j(a;x)<rg; (3) D r [a℄=fx2X j(a;x)rg; (4) S r (a)=fx2X j(a;x)=rg (5)
are alled, respe tively, open ball, losed ball, and sphere of the spa e
(X;) with enter at a and radius r.
Subspa es of a Metri Spa e
If (X;) is a metri spa e and A X, then the restri tion of metri
to AA is ametri inA, and (A;
AA
) is a metri spa e. It is alled
a subspa e of (X;). The ballD 1 [0℄ and sphere S 1 (0) in R n
(with Eu lidean metri , see 3.C)
are denoted by symbolsD n
and S n 1
and alled n-dimensionalball and
(n 1)-dimensional sphere. They are onsidered asmetri spa es (with
the metri restri ted fromR n ). 3.D. Che k that D 1 is the segment [ 1;1℄; D 2 is a disk; S 0 is the pair of pointsf 1;1g;S 1 is a ir le; S 2 is asphere; D 3 isa ball.
The last two statements larify the origin of terms sphere and ball (in
the ontext of metri spa es).
Some properties of balls and spheres in arbitrary metri spa e
resem-ble familiar properties of planar disks and ir les and spatial balls and
spheres.
3.E. Prove that for points x and a of any metri spa e and any r >
(a;x) D r (a;x) (x)D r (a): Surprising Balls
Howeverin othermetri spa esballsandspheresmayhaverathersurprising
properties.
3.6. WhatareballsandspheresinR 2
withmetri sof3.1and3.2( f. 3.4)?
3.7. Find D 1 [a℄,D1 2 [a℄,andS1 2
(a)in thespa eof3.A.
3.8. Find a metri spa e and two balls in it su h that the ball with the
3.9. Whatistheminimalnumberofpointsinthespa ewhi hisrequiredto
be onstru tedin 3.8.
3.10. Provethat in 3.8 the big radius does not ex eed double the smaller
radius.
Segments (What Is Between)
3.11. Provethatthesegmentwithend pointsa;b2R n
anbedes ribedas
fx2R n
j(a;x)+(x;b)=(a;b)g;
whereistheEu lideanmetri .
3.12. How do the sets de ned as in 3.11 look like with of 3.1 and 3.2?
(Consider the asen=2ifitappearstobeeasier.)
Bounded Sets and Balls
A subset A of a metri spa e (X;) is said to be bounded, if there is a
numberd>0su h that (x;y)<d forany x;y2A. The greatest lower
bound of su h d is alled the diameter of A and denoted by diam(A).
3.F. Prove that a set A is bounded, i itis ontained in aball.
3.13. What is the relation between the minimal radius of su h a ball and
diam(A)?
Norms and Normed Spa es
Let X be ave torspa e (overR). Fun tionX !R
+
: x7!jjxjj is alled a
norm if
(a) jjxjj=0,i x=0;
(b) jjxjj=jjjjxjjforany2R andx2X;
( ) jjx+yjjjjxjj+jjyjjforanyx;y2X.
3.14. Provethatifx7!jjxjjisanormthen
:XX!R
+
:(x;y)7!jjx yjj
isametri .
The ve torspa e equipped with aspe i ed norm is alled a normedspa e.
Themetri de ned by thenormasin 3.14 turns thenormedspa e intothe
metri onein a anoni alway.
3.15. Lookthroughtheproblemsofthisse tionand gureoutwhi hofthe
metri spa esinvolvedare,in fa t,normedve torspa es.
3.16. Provethateveryballinthenormedspa eisa onvex 3
setsymmetri
withrespe tto the enteroftheball.
3
Re all thataset Ais saidto be onvex iffor anyx;y 2Athesegment onne ting
x;y is ontained in A. Of ourse, this de nition is basedonthe notionof segment,
3.17*. Provethatevery onvex losedboundedsetinR n
,whi his
symmet-ri withrespe t toits enter andisnot ontainedin anyaÆnespa eex ept
R n
itself, isthe unit ball withrespe tto somenorm, and that this normis
uniquelyde nedbythisball.
Metri Topology
3.G. The olle tion of all open balls in the metri spa e is a base for
some topology ( f.2.A, 2.Band 3.E).
This topology is alled metri topology. It is said to be indu ed by the
metri . This topologi al stru ture is always meantwhenever the metri
spa e is onsidered as a topologi al one (for instan e, when one says
about open and losed sets, neighborhoods, et . in this spa e).
3.H. Prove that the standard topologi al stru ture in R introdu ed in
Se tion 1 isindu ed by metri (x;y)7!jx yj.
3.18. Whattopologi alstru tureisindu edbythemetri of3.A?
3.I. A set isopen ina metri spa e, i it ontains together with any its
pointa ballwith enter at this point.
3.19. Provethata losedballis losed(withrespe ttothemetri topology).
3.20. Finda losedball,whi hisopen(withrespe ttothemetri topology).
3.21. Find anopen ball, whi h is losed(with respe tto the metri
topol-ogy).
3.22. Provethatasphereis losed.
3.23. Findasphere,whi hisopen.
Metrizable Topologi al Spa es
A topologi alspa e issaid tobemetrizable if its topologi alstru tureis
indu ed by some metri .
3.J. An indis rete spa e is not metrizable unless it onsists of a single
point(it has too fewopen sets).
3.K. A nite spa eis metrizable i itis dis rete.
3.24. Whi htopologi alspa esdes ribedinSe tion 1aremetrizable?
Equivalent Metri s
3.25. Arethemetri sof3.C,3.1,and3.2equivalent?
3.26. Prove that metri s
1 ,
2
in X are equivalent if there are numbers
;C>0su hthat 1 (x;y) 2 (x;y)C 1 (x;y) foranyx;y2X.
3.27. Generallyspeakingtheinverseis nottrue.
3.28. Riddle. Hen ethe onditionoftheequivalen eofmetri sformulated
in 3.26 an beweakened. How?
3.29*. Provethatthefollowingtwometri s
1 ,
C
inthesetofall
ontin-uousfun tions [0;1℄!R arenotequivalent: 4 1 (f;g)= Z 1 0 f(x) g(x) dx; C (f;g)= max x2[0;1℄ f(x) g(x) :
Is it true that topologi al stru ture de ned by one of them is ner than
another?
Ultrametri
A metri is alled an ultrametri if it satis es to ultrametri triangle
in-equality:
(x;y)maxf(x;z);(z;y)g
foranyx;y,z.
A metri spa e(X;)withultrametri is alledanultrametri spa e.
3.30. Che kthatonlyonemetri in3.A{3.2 isultrametri . Whi hone?
3.31. Provethat inanultrametri spa e alltrianglesare isos eles(i.e., for
anythree pointsa, b, twoofthethree distan es (a;b), (b; ), (a; ) are
equal).
3.32. Provethatinaultrametri spa espheresarenotonly losed( f.3.22)
but alsoopen.
Themostimportantexampleof ultrametri isp-adi metri inthe setQ of
all rational numbers. Let pbe aprime number. For x;y 2 Q, presentthe
di eren ex y as r
s p
,wherer,s,and areintegers,andr,sarerelatively
prime withp. Put(x;y)=p
.
3.33. Provethatthis isanultrametri .
Operations with Metri s
3.34. Provethatif:XX!R
+
isafun tionwhi hsatis es onditions
(a)and( ) ofthede nition ofmetri thenthefun tion
(x;y)7!(x;y)+(y;x)
isametri in X.
3.35. Provethatif
1 ,
2
aremetri sin X then
1 + 2 andmaxf 1 ; 2 gare
alsometri s. Arethefun tionsminf
1 ; 2 g, 1 2 ,and 1 2 metri s? 3.36. Provethatif:XX !R + isametri then (a) fun tion (x;y)7! (x;y) 1+(x;y) isametri ; (b) fun tion (x;y)7!f (x;y)
isametri ,iff satis esthefollowing onditions:
(1) f(0)=0,
(2) f isamonotonein reasingfun tion,and
(3) f(x+y)f(x)+f(y)foranyx;y2R.
3.37. Provethatmetri sand
1+
areequivalent.
Distan e Between Point and Set
Let (X;) be ametri spa e, AX,b 2X. The inff(b;a)ja2Agis
alled a distan e from the point b to the set A and denoted by (b;A).
3.L. Let A bea losed set. Prove that (b;A)=0, i b 2A.
3.38. Provethat j(x;A) (y;A)j(x;y)forany setA andpointsx,y
ofthesamemetri spa e.
Distan e Between Sets
LetAandB bebounded subsetsin themetri spa e(X;). Put
d (A;B)=max n sup a2A (a;B);sup b2B (b;A) o :
Thisnumberis alledtheHausdor distan e betweenAandB.
3.39. ProvethattheHausdor distan einthesetofallbounded subsetsof
ametri spa esatis esthe onditions(b)and( )ofthede nition ofmetri .
3.40. Provethat for everymetri spa e theHausdor distan e is a metri
in thesetofits losedbounded subsets.
LetAandB bebounded polygonsin theplane 5
. Put
d
(A;B)=S(A)+S(B) 2S(A\B);
where S(C)istheareaofpolygonC.
3.41. Provethatd
isametri inthesetofallplanebounded polygons.
5
Althoughweassumethatthenotionofboundedpolygoniswell-knownfrom
elemen-tary geometry, re allthede nition. A bounded planepolygonis aset ofthepoints
ofasimple losedpolygonallineandthepointssurroundedbythisline. Byasimple
losed polygonalline wemeana y li sequen e ofsegmentssu hthat ea h ofthem
Wewill all d
theareametri .
3.42. Provethatinthesetofallboundedplanepolygonstheareametri is
notequivalentto theHausdor metri .
3.43. Provethatinthesetof onvexboundedplanepolygonstheareametri
isequivalentto theHausdor metri .
4. Subspa es
Let(X;) be atopologi alspa e,and AX. Denote by
A
the
olle -tion of sets A\V, where V 2.
4.A.
A
is atopologi alstru ture inA.
The pair (A;
A
)is alledasubspa e of the spa e (X;). The olle tion
A
is alledthesubspa etopology ortherelativetopology orthetopology
indu ed on Aby , and its elements are alled open sets inA.
4.B. The anoni al topologyin R 1
and the topology indu ed on R 1
as
a subspa e ofR 2
oin ide.
4.1. Riddle. Howto onstru tabaseforthetopologyindu edonAusing
thebaseforthetopologyin X?
4.2. Des ribethetopologi alstru turesindu ed
(a) onthesetN ofnaturalnumbersbythetopologyoftherealline;
(b) onN bythetopologyofthearrow;
( ) onthetwo-pointsetf1;2gbythetopologyofR
T
1 ;
(d) onthesameset bythetopologyofthearrow.
4.3. Is thehalf-openinterval[0;1) open in thesegment [0;2℄ onsidered as
asubspa eoftherealline?
4.C. Asetis losedinasubspa e,i itistheinterse tionofthesubspa e
and a losed subset of the ambientspa e.
Relativityof Openness
Sets, whi h are open in the subspa e, are not ne essarily open in the
ambient spa e.
4.D. The unique open set inR 1
, whi h isalsoopen inR 2
, isthe empty
set ?.
However:
4.E. Open sets of anopen subspa e are open inthe ambientspa e, i.e.,
if A2then
A .
4.F. Closedsets of the losedsubspa e are losedinthe ambientspa e.
4.4. ProvethatasetU isopenin X,i everyitspointhasaneighborhood
V in X su hthatU \V isopeninV.
Itallowsoneto saythat thepropertyofbeingopenisalo alproperty.
4.5. Showthatthepropertyofbeing losedisnotalo alproperty.
4.G Transitivity of Indu ed Topology. Let (X;) be a topologi al
spa e,and X AB. Then (
A )
B
=
B
, i.e.,the topology indu edon
B by thetopologyindu ed onA oin ides with thetopologyindu edon B
dire tly.
4.6. Let (X;) be a metri spa e, and A X. Then the topology in A
generated bymetri
AA
oin ideswiththetopologyindu edonA bythe
topologyin X generatedbymetri . (Toprovethis statementyouneedto
provetwoin lusions. Whi hofthemislessobvious?)
Agreement on Notations of Topologi al Spa es
Di erenttopologi alstru tures inthesame setare not onsidered
simul-taneously very often. That iswhy atopologi alspa e isusually denoted
by the same symbol as the set of its points, i.e., instead of (X;) one
writes just X. The same is applied for metri spa es: instead of (X;)
one writes justX.
5. Position of a Point with Respe t to a Set
This se tionis devoted toa further expansion of the vo abulary needed
when one speaksof phenomena ina topologi alspa e.
Interior, Exterior and Boundary Points
Let X be atopologi al spa e,A X,and b 2X. The pointb is alled
an interior point of the set A if it has a neighborhood ontained
in A;
an exterior pointof the set A ifithas aneighborhooddisjointwith
A;
Interior and Exterior
The interior of a set A in a topologi al spa e X is the maximal (with
respe t to in lusion) open in X set ontained in A, i.e., an open set,
whi h ontains any other open subset of A. It is denoted IntA or,going
intodetails, Int
X A.
5.A. Every subsetof a topologi alspa e has interior. It is the union of
all open sets ontained in this set.
5.B. The interiorof a set is the union of itsinteriorpoints.
5.C. A set is open, i it oin ides with itsinterior.
5.D. Prove that inR:
(a) Int [0;1)=(0;1),
(b) IntQ =? and
( ) Int (R rQ)=?.
5.1. Find theinterioroffa;b;dgin spa e4pT.
The exterior of a set is the maximal open set disjoint from A. It is
obvious that the exterior of A is Int (X rA).
Closure
The losure of a set A is the minimal losed set ontaining A. It is
denoted ClA or, going intodetails, Cl
X A.
5.E. Everysubsetof topologi alspa e has losure. Itisthe interse tion
of all losed sets ontainingthis set.
5.2. ProvethatifAisasubspa eofX,andBA,thenCl
A
B=(Cl
X B)\
A. IsittruethatInt
A
B=(Int
X
B)\A?
A pointbis alledanadherentpoint foraset Aifallofitsneighborhood
interse tA.
5.F. The losureof a set is the set of itsadherent points.
5.G. A set A is losed,i A=ClA.
5.H. The losureof asetis the omplementofitsexterior. In formulas:
ClA=XrInt (X rA), whereX is the spa e and AX.
5.I. Provethat inR:
(a) Cl[0;1)=[0;1℄,
(b) ClQ =R,
Frontier
The frontier of a set A is the set ClArIntA. It is denoted by FrA or,
more pre isely, Fr
X A.
5.4. In4pT ndthefrontieroffag.
5.J. The frontier of a set is the set of its boundary points.
5.K. Prove that a set A is losed, i FrA A.
5.5. Provethat FrA =Fr(XrA). Find aformula forFrA, whi his
sym-metri withrespe ttoAand XrA.
5.6. Thefrontierof aset A equals theinterse tion of the losure ofA and
the losureofthe omplementofA:
FrA=ClA\Cl (XrA):
Closure and Interior with Respe t to a Finer Topology
5.7. Let
1 ,
2
betopologi alstru tureinX,and
1 2 . LetCl i denote
the losurewithrespe tto
i . Provethat Cl 1 ACl 2 AforanyAX.
5.8. Formulateandproveananalogousstatementaboutinterior.
Properties of Interior and Closure
5.9. Provethat ifAB thenIntAIntB.
5.10. ProvethatIntIntA=IntA.
5.11. IsittruethatforanysetsAandB thefollowingequalitiesholdtrue:
Int(A\B)=IntA\IntB; (6)
Int(A[B)=IntA[IntB? (7)
5.12. Giveanexamplein whi honeofthatequalitiesdoesnothold true.
5.13. Intheexamplethat you havefoundsolvingthepreviousprobleman
in lusion of one hand side into another one holds true. Does this in lusion
holdtrueforanyAandB?
5.14. StudytheoperatorCl in away suggestedbytheinvestigation ofInt
undertakenin 5.9{5.13.
5.15. FindClf1g,Int[0;1℄,andFr(2;+1) inthearrow.
5.16. FindInt (0;1℄[f2g ,Cl f 1 n jn2Ng ,andFrQ inR.
5.17. Find ClN, Int (0;1), andFr[0;1℄in R
T1
. Howto ndthe losureand
interiorofasetinthisspa e?
5.18. Prove that a sphere ontains the frontier of the open ball with the
same enterandradius.
Let A be a subset, and b be a point of the metri spa e (X;). Re all
(see Se tion3) that the distan e (b;A)from the point b to the set A is
the inff(b;a)ja2Ag.
5.L. Prove that b2ClA,i (b;A)=0.
5.20 The Kuratowski Problem. Howmanypairwisedistin tsets anone
obtainoutofasingleset usingoperatorsClandInt?
Thefollowingproblems willhelp you to solve problem 5.20.
5.20.1. Find a set A R su h that the sets A, ClA, and IntA
wouldbe pairwisedistin t.
5.20.2. Is there aset AR su hthat
(a) A,ClA,IntA,ClIntA arepairwisedistin t;
(b) A,ClA,IntA,IntClA arepairwisedistin t;
( ) A,ClA,IntA,ClIntA,IntClA arepairwisedistin t?
Ifyou ndsu hsets,keepongoinginthesameway,andwhenfail,
try to formulate a theorem explainingthefailure.
5.20.3. Prove that ClIntClIntA=ClIntA.
5.21*. Find threesetsin thereal line,whi hhavethesamefrontier. Isit
possibleto in reasethenumberof su hsets?
Re allthatasetAR n
issaidtobe onvex iftogetherwithanytwopoints
it ontains the whole interval onne ting them (i.e., for any x;y 2 A any
pointz belongingto thesegment[x;y℄belongsto A).
LetAbea onvexsetin R n
.
5.22. ProvethatClAandIntAare onvex.
5.23. ProvethatA ontainsaball,unlessAisnot ontainedinan(n
1)-dimensional aÆnesubspa eofR n
.
5.24. WhenisFrA onvex?
Chara terization of Topology by Closure or Interior
Opera-tions
5.25*. Letin theset of allsubsetof aset X exist anoperator Cl
whi h
hasthefollowingproperties:
(a) Cl ?=?; (b) Cl AA; ( ) Cl (A[B)=Cl A[Cl B; (d) Cl Cl A=Cl A. Provethat=fU X jCl (XrU)=XrUgisatopologi alstru ture, andCl
Dense Sets
Let A and B be sets in a topologi al spa e X. A is said to be dense in
B if ClAB, and everywhere dense if ClA=X.
5.M. Aset iseverywhere dense,i itinterse ts any nonempty open set.
5.N. The set Q is everywhere dense inR.
5.27. Givea hara terizationofeverywheredensesetsinanindis retespa e,
in thearrowandin R
T1 .
5.28. Provethat atopologi al spa eis adis rete spa e, i it hasa unique
everywheredenseset(whi h istheentirespa e,of ourse).
5.29. Isittruethattheunionofeverywheredensesetsiseverywheredense,
andthat theinterse tion ofeverywheredensesetsiseverywheredense?
5.30. Provethat the interse tion of twoopen everywheredense sets is
ev-erywheredense.
5.31. Whi h onditioninthepreviousproblem isredundant?
5.32*. Provethat in R a ountableinterse tion ofopen everywheredense
sets is everywhere dense. Is it possible to repla e R here by an arbitrary
topologi alspa e?
5.33*. Prove that Q annot be presented as a ountable interse tion of
opensets densein R.
5.34. Formulate a ne essaryand suÆ ient ondition on the topology of a
spa e whi h hasaneverywheredensepoint. Find spa essatisfyingthe
on-dition inSe tion1.
Nowhere Dense Sets
A set is allednowhere dense if its exterioris everywhere dense.
5.35. Canasetbeeverywheredenseandnowheredensesimultaneously?
5.O. A set A isnowhere dense in X,i any neighborhoodof any point
x 2 X ontains a point y su h that the omplement of A ontains y
together with one of itsneighborhoods.
5.36. Riddle. What anyousayabouttheinteriorofanowheredenseset?
5.37. IsR nowheredensein R 2
?
5.38. ProvethatifAisnowheredensethenIntClA=?.
5.39. Provethat the frontier ofa losed set isnowhere dense. Is this true
fortheboundaryofanopenset; boundaryofanarbitraryset?
5.40. Provethata niteunionofnowheredensesetsisnowheredense.
5.41. ProvethatinR n
(n1)everyproperalgebrai set(i.e.,asetde ned
byalgebrai equations)isnowhere dense.
5.42. ProvethatforeverysetAthereexistsamaximalopensetB inwhi h
Limit Points and Isolated Points
A point b is alled a limit point of a set A if any neighborhood of b
interse ts Arfbg.
5.P. Every limitpoint of a set is itsadherent point.
5.43. Give anexample provingthat an adherent point may be notalimit
one.
A point b is alled an isolated point of a set A if b 2 A and there exists
a neighborhoodof b disjointwith Arfbg.
5.Q. A set A is losed, i it ontains allits limitpoints.
5.44. Find limit and isolated points of the sets (0;1℄[f2g, f 1
n
j n 2 Ng
in Q andin R.
5.45. FindlimitandisolatedpointsofthesetN inR
T1 .
Lo ally Closed Sets
AsubsetAofatopologi alspa eXis alledlo ally losedifea hofitspoints
hasaneighborhoodU su hthatA\U is losedinU ( f.4.4{4.5).
5.46. Provethatthefollowing onditionsareequivalent:
(a) Ais lo ally losedinX;
(b) Ais anopensubsetofits losureCl
X A;
( ) Ais theinterse tionof openand losedsubsetsof X.
6. Set-Theoreti Digression. Maps
Maps and the Main Classes of Maps
A mapping f of a set X to a set Y is a triple onsisting of X, Y, and
a rule, 6
whi h assigns to every element of X exa tly one element of Y.
There are other words with the same meaning: map, fun tion.
Iff isamappingofX toY then onewritesf :X !Y, orX f
!Y. The
element b of Y assigned by f to an element a of X is denoted by f(a)
and alled the image of a under f. One writes b = f(a), or a f
7! b, or
f :a7!b.
A mappingf : X !Y is alled a surje tive map, or just a surje tion if
everyelementof Y is animageof atleast oneelementof X. A mapping
6
Of ourse, the rule (as everything in the set theory) may be thought of as a set.
Namely, one onsiders asetof orderedpairs(x;y)withx 2X,y 2Y su h thatthe
f : X ! Y is alled an inje tive map, inje tion, or one-to-one map if
every element of Y is an image of not more than one element of X. A
mappingis alledabije tive map,bije tion,orinvertibleifitissurje tive
and inje tive.
Image and Preimage
The image of a set A X under a map f :X !Y is the set of images
of all points of A. It is denoted by f(A). Thus
f(A)=ff(x) : x2Ag:
The imageof the entire set X (i.e., f(X)) is alled the image of f. The
preimage of a set B Y under a map f :X !Y is the set of elements
of X whose images belong toB. It is denoted by f 1
(B). Thus
f 1
(B)=fa2X : f(a)2Bg:
Be areful with these terms: their etymology an be misleading. For
example, the image of the preimage of a set B an di er from B. And
even if it does not di er, It may happen that the preimage is not the
only set with this property. Hen e, the preimage annot bede ned asa
set whose imageis a given set.
6.A. f f 1
(B)
=B, i B is ontainedin the image of f.
6.B. f f 1
(B)
B forany map f :X !Y and B Y.
6.C. Let f :X ! Y and B Y su h that f f 1
(B)
= B. Then the
following statements are equivalent:
(a) f 1
(B) is the unique subset of X whoseimage equals B;
(b) for any a 1 ;a 2 2f 1 (B) the equality f(a 1 )=f(a 2 ) implies a 1 =a 2 .
6.D. A map f : X ! Y is an inje tion, i for any B Y su h that
f f 1 (B) = B the preimage f 1
(B) is the unique subset of X whose
image equals B.
6.E. f 1
f(A)
A for any mapf :X !Y and AX.
6.F. f 1
f(A)
=A, i f(A)\f(X rA)=?.
6.1. DothefollowingequalitiesholdtrueforanyA;BY andanyf :X !
Y: f 1 (A[B)=f 1 (A)[f 1 (B); (8) f 1 (A\B)=f 1 (A)\f 1 (B); (9) f 1 (Y rA)=Xrf 1 (A)? (10)
6.2. DothefollowingequalitiesholdtrueforanyA;B X andanyf :X ! Y: f(A[B)=f(A)[f(B); (11) f(A\B)=f(A)\f(B); (12) f(XrA)=Y rf(A)? (13)
6.3. Giveexamplesin whi h twooftheequalitiesabovearefalse.
6.4. Repla ethefalseequalitiesof6.2by orre tin lusions.
6.5. What simple onditionon f : X ! Y should be imposed in order to
make orre talltheequalitiesof6.2foranyA;BX ?
6.6. Provethat foranymapf :X !Y,andsubsetsAX,BY:
B\f(A)=f f 1
(B)\A
:
Identity and In lusion
The identity map of a set X is the map X ! X de ned by formula
x 7! x. It is denoted by id
X
, or just id, when there is no ambiguity. If
A is a subset of X then the map A ! X de ned by formula x 7! x is
alled an in lusion map, or just in lusion, of A into X and denoted by
in:A!X, orjust in,when A and X are lear.
6.G. The preimage of a set B under anin lusionin:A!X isB \A.
Composition
The omposition of mappings f :X !Y and g :Y !Z isthe mapping
gÆf :X !Z de ned by formulax7!g f(x)
.
6.H. hÆ(gÆf)=(hÆg)Æf for any maps f :X !Y, g :Y !Z, and
h :Z !U. 6.I. f Æ(id X )=f =(id X )Æf forany f :X !Y.
6.J. The omposition of inje tionsis inje tive.
6.K. Ifthe omposition gÆf is inje tivethen f isinje tive.
6.L. The ompositionof surje tions issurje tive.
6.M. Ifthe omposition gÆf is surje tive theng is surje tive.
Inverse and Invertible
Amapg :Y !X issaidtobeinverse toamapf :X !Y ifgÆf =id
X
and f Æg =id
Y
. A map, for whi h an inverse map exists, is said to be
invertible.
6.O. A mappingis invertible,i it isa bije tion.
6.P. Ifan inverse map exists then it is unique.
Submappings
If A X and B Y then for every f : X ! Y su h that f(A) B
there ismappingab(f):A!B de ned by formulax7!f(x)and alled
an abbreviation ofthe mappingf toA;B, orsubmapping, orsubmap. If
B =Y thenabf :A!Y isdenoted by f
A
and alledthe restri tion of
f toA. IfB 6=Y then abf :A !B isdenoted by f
A;B
or even simply
fj.
6.Q. The restri tion of a map f :X !Y to AX is the omposition
of in lusion inA :!X and f. In other words, f
A
=fÆin.
6.R. Any abbreviation (in luding any restri tion) of inje tions is
inje -tive.
6.S. Ifarestri tionofamappingissurje tivethentheoriginalmapping
is surje tive.
7. Continuous Maps
De nition and Main Properties of Continuous Maps
Let X, Y be topologi al spa es. A map f : X ! Y is said to be
ontinuous if the preimage of any open subset of Y isan open subset of
X.
7.A. A map is ontinuous, i the preimage of any losed set is losed.
7.B. The identity map of any topologi al spa eis ontinuous.
7.1. Let
1 ,
2
be topologi al stru tures in X. Prove that the identity
mappingofX id:(X; 1 )!(X; 2 ) is ontinuous,i 2 1 .
7.2. Letf :X !Y bea ontinuousmap. Isit ontinuouswithrespe tto
(a) a nertopologyin X andthesametopologyinY,
(b) a oarsertopologyinX andthesametopologyinY,
7.3. Let X be a dis rete spa e and Y an arbitrary spa e. Whi h maps
X !Y andY !X are ontinuous?
7.4. Let X bean indis rete spa e and Y anarbitrary spa e. Whi h maps
X !Y andY !X are ontinuous?
7.C. LetAbeasubspa eofX. Thein lusionin:A!X is ontinuous.
7.D. The topology
A
indu ed on A X by the topology of X is the
oarsest topology in A su h that the in lusion mapping in : A ! X is
ontinuous with respe t to it.
7.5. Riddle. The statement 7.D admits anatural generalizationwith the
in lusion map repla ed byan arbitrarymap f : A!X of an arbitraryset
A. Find thisgeneralization.
7.E. A ompositionof ontinuous maps is ontinuous.
7.F. A submapof a ontinuous map is ontinuous.
7.G. A map f :X ! Y is ontinuous, i abf : X !f(X) is
ontinu-ous.
7.H. Any onstant map (i.e., a map with image onsisting of a single
point)is ontinuous.
Reformulations of De nition
7.6. Provethat amappingf :X !Y is ontinuous,i
Clf 1 (A)f 1 (ClA) foranyAY.
7.7. Formulateandprovesimilar riteriaof ontinuityintermsofIntf 1
(A)
andf 1
(IntA). DothesameforClf(A)andf(ClA).
7.8. Let be a base for topology in Y. Provethat amap f : X ! Y is
ontinuous,i f 1
(U)is openforanyU 2.
More Examples
7.9. Isthemappingf :[0;2℄