Elementary Topology in Problems

5/21/2018 Elementary Topology in Problems

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Elementary TopologyA First Course

Textbook in Problems

O. Y. Viro, O. A. Ivanov,

N. Y. Netsvetaev, V. M. Kharlamov


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This book includes basic material on general topology, introducesalgebraic topology via the fundamental group and covering spaces,and provides a background on topological and smooth manifolds. Itis written mainly for students with a limited experience in mathe-matics, but determined to study the subject actively. The materialis presented in a concise form, proofs are omitted. Theorems, how-ever, are formulated in detail, and the reader is expected to treatthem as problems.


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Foreword

Genre, Contents and Style of the Book

The core of the book is the material usually included in the Topol-ogy part of the two year Geometry lecture course at the MathematicalDepartment of St. Petersburg University. It was composed by VladimirAbramovich Rokhlin in the sixties and has almost not changed sincethen.

We believe this is the minimum topology that must be mastered byany student who has decided to become a mathematician. Studentswith research interests in topology and related fields will surely needto go beyond this book, but it may serve as a starting point. The bookincludes basic material on general topology, introduces algebraic topologyvia its most classical and elementary part, the theory of the fundamentalgroup and covering spaces, and provides a background on topologicaland smooth manifolds. It is written mainly for students with a limitedexperience in mathematics, but who are determined to study the subjectactively.

The core material is presented in a concise form; proofs are omit-ted. Theorems, however, are formulated in detail. We present them asproblems and expect the reader to treat them as problems. Most of thetheorems are easy to find elsewhere with complete proofs. We believethat a serious attempt to prove a theorem must be the first reaction toits formulation. It should precede looking for a book where the theoremis proved.

On the other hand, we want to emphasize the role of formulations.In the early stages of studying mathematics it is especially important totake each formulation seriously. We intentionally force a reader to thinkabout each simple statement. We hope that this will make the bookinconvenient for mere skimming.

The core material is enhanced by many problems of various sortsand additional pieces of theory. Although they are closely related to themain material, they can be (and usually are) kept outside of the standardlecture course. These enhancements can be recognized by wider margins,

as the next paragraph.

iii


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FOREWORD iv

The problems, which do not comprise separate topics and are intendedexclusively to be exercises, are typeset with small face. Some of them are

very easy and included just to provide additional examples. Few problemsare difficult. They are to indicate relations with other parts of mathematics,show possible directions of development of the subject, or just satisfy anambitious reader. Problems, whose solutions seem to be the most difficult(from the authors viewpoint), are marked with a star, as in many otherbooks.

Further, we want to deliver additional pieces of theory (with respect to the corematerial) to more motivated and advanced students. Maybe, a mathematician, whodoes not work in the fields geometric in flavor, can afford the luxury not to knowsome of these things. Maybe, students studying topology can postpone this materialto their graduate study. We would like to include this in graduate lecture courses.

However, quite often it does not happen, because most of the topics of this sort arerather isolated from the contents of traditional graduate courses. They are important,but more related to the material of the very first topology course. In the book thesetopics are intertwined with the core material and exercises, but are distinguishable:they are typeset, like these lines, with large face, theorems and problems in them arenumerated in a special manner described below.

Exercises and illustrative problems to the additional topics are typesetwith wider margins and marked in a different way.

Thus, the whole book contains four layers:

the core material,

exercises and illustrative problems to the core material, additional topics, exercises and illustrative problems to additional topics.

The text of the core material is typeset with large face and smallestmargins.

The text of problems elaborating on the core material is typeset withsmall face and larger margins.

The text of additional topics is typeset is typeset with large face as the problemselaborating on the core material.

The text of problems illustrating additional topics is typeset with smallface and larger margins.

Therefore the book looks like a Russian folklore doll, matreshkacom-posed of several dolls sitting inside each other. We apologize for beingnonconventional in this and hope that it may help some readers and doesnot irritate the others too much.

The whole text of the book is divided into sections. Each section isdivided into subsections. Each of them is devoted to a single topic andconsists of definitions, commentaries, theorems, exercises, problems, and

riddles.


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FOREWORD v

By a riddlewe mean a problem of a special sort: its solution is notcontained in the formulation. One has to guess a solution, rather thandeduce it.

0.A. Theorems, exercises, problems and riddles belonging to the corematerial are marked with pairs consisting of the number of section anda letter separated with a dot. The letter identifies the item inside thesection.

0.1. Exercises, problems, and riddles, which are not included in the core, butare closely related to it (and typeset with small face) are marked with pairsconsisting of the number of the section and the number of the item inside thesection. The numbers in the pair are separated also by a dot.

Theorems, exercises, problems and riddles related to additional topicsare enumerated independently inside each section and denoted similarly.

0:A. The only difference is that the components of pairs marking the items areseparated by a colon (rather than dot).

We assume that the reader is familiar with naive set theory, butanticipate that this familiarity may be superficial. Therefore at pointswhere set theory is especially crucial we make set-theoretic digressionsmaintained in the same style as the rest of the book.

Advices to the Reader

Since the book contains a summary of elementary topology, you mayuse the book while preparing for an examination (especially, if the examreduces to solving a collection of problems). However, if you attendlectures on the subject, it would be much wiser to read the book priorto the lectures and prove theorems before the lecturer gives the proofs.

We think that a reader who is able to prove statements of the coreof the book, does not need to solve all the other problems. It would bereasonable instead to look through formulations and concentrate on the

most difficult problems. The more difficult the theorems of the main textseem to you, the more carefully you should consider illustrative problems,and the less time you should waste with problems marked with stars.

Keep in mind that sometimes a problem which seems to be difficult isfollowed by easier problems, which may suggest hints or serve as technicallemmas. A chain of problems of this sort is often concluded with aproblem which suggests a return to the theorem, once you are armedwith the lemmas.

Most of our illustrative problems are easy to invent, and, moreover, ifyou study the subject seriously, it is always worthwhile to invent problems

of this sort. To develop this style of studying mathematics while solving


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FOREWORD vi

our problems one should attempt to invent ones own problems and solvethem (it does not matter if they are similar to ours or not). Of course,some problems presented in this book are not easy to invent.


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Contents

Foreword iiiGenre, Contents and Style of the Book iiiAdvices to the Reader v

Part 1. General Topology 1

Chapter 1. Generalities 3

1. Digression on Sets 311 Sets and Elements 312 Equality of Sets 413 The Empty Set 414 Basic Sets of Numbers 515 Describing a Set by Listing of Its Elements 516 Subsets 6

17 Properties of Inclusion 6

18 To Prove Equality of Sets, Prove Inclusions 619 Inclusion Versus Belonging 6110 Defining a Set by a Condition 7111 Intersection and Union 7112 Different Differences 9Proofs and Comments 10Hints, Comments, Advises, Solutions, and Answers 11

2. Topology in a Set 1221 Definition of Topological Space 1222 The Simplest Examples 122

3 The Most Important Example: Real Line 1324 Additional Examples 1325 Using New Words: Points, Open and Closed Sets 1326 Set-Theoretic Digression. De Morgan Formulas 1427 Properties of Closed Sets 1428 Being Open or Closed 1429 Cantor Set 15210 Characterization of Topology in Terms of Closed Sets 15211 Topology and Arithmetic Progressions 15212 Neighborhoods 16Proofs and Comments 16

Hints, Comments, Advises, Solutions, and Answers 17

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CONTENTS viii

3. Bases 19

31 Definition of Base 19

32 When a Collection of Sets is a Base 1933 Bases for Plane 1934 Subbases 2035 Infiniteness of the Set of Prime Numbers 2036 Hierarchy of Topologies 20Proofs and Comments 21Hints, Comments, Advises, Solutions, and Answers 21

4. Metric Spaces 2341 Definition and First Examples 2342 Further Examples 234

3 Balls and Spheres 2444 Subspaces of a Metric Space 2445 Surprising Balls 2546 Segments (What Is Between) 2547 Bounded Sets and Balls 2548 Norms and Normed Spaces 2549 Metric Topology 26410 Openness and Closedness of Balls and Spheres 26411 Metrizable Topological Spaces 27412 Equivalent Metrics 27

413 Ultrametric 27

414 Operations with Metrics 28415 Distance Between Point and Set 28416 Distance Between Sets 28417 Asymmetrics 29Proofs and Comments 30Hints, Comments, Advises, Solutions, and Answers 31

5. Ordered Sets 3451 Strict Orders 3452 Non-Strict Orders 3453 Relation between Strict and Non-Strict Orders 355

4 Cones 3555 Position of an Element with Respect to a Set 3656 Total Orders 3757 Topologies Defined by a Total Order 3758 Poset Topology 3859 How to Draw a Poset 39510 Cyclic Orders in Finite Set 40511 Cyclic Orders in Infinite Sets 41512 Topology of Cyclic Order 42Proofs and Comments 42Hints, Comments, Advises, Solutions, and Answers 43


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CONTENTS ix

6. Subspaces 45

61 Topology for a subset of a space 45

62 Relativity of Openness 4563 Agreement on Notations of Topological Spaces 46Proofs and Comments 46Hints, Comments, Advises, Solutions, and Answers 47

7. Position of a Point with Respect to a Set 4871 Interior, Exterior and Boundary Points 4872 Interior and Exterior 4873 Closure 4874 Closure in a Metric Space 4975 Frontier 497

6 Closure and Interior with Respect to a Finer Topology 4977 Properties of Interior and Closure 5078 Compositions of Closure and Interior 5079 Sets with Common Frontier 51710 Convexity and Int, Cl, Fr 51711 Characterization of Topology by Closure or Interior

Operations 51712 Dense Sets 51713 Nowhere Dense Sets 52714 Limit Points and Isolated Points 52

715 Locally Closed Sets 53

Proofs and Comments 53Hints, Comments, Advises, Solutions, and Answers 53

8. Set-Theoretic Digression. Maps 5581 Maps and the Main Classes of Maps 5582 Image and Preimage 5583 Identity and Inclusion 5684 Composition 5685 Inverse and Invertible 5786 Submappings 57

9. Continuous Maps 58

91 Definition and Main Properties of Continuous Maps 5892 Reformulations of Definition 5993 More Examples 5994 Behavior of Dense Sets 5995 Local Continuity 6096 Properties of Continuous Functions 6097 Continuity of Distances 6198 Isometry 6199 Gromov-Hausdorff distance 61910 Contractive maps 62

911 Monotone maps 62


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CONTENTS x

912 Functions on Cantor Set and Square-Filling Curves 62

913 Sets Defined by Systems of Equations and Inequalities 63

914 Set-Theoretic Digression. Covers 64915 Fundamental Covers 64Hints, Comments, Advises, Solutions, and Answers 65

10. Homeomorphisms 67101 Definition and Main Properties of Homeomorphisms 67102 Homeomorphic Spaces 67103 Role of Homeomorphisms 67104 More Examples of Homeomorphisms 68105 Examples of Homeomorphic Spaces 69106 Examples of Nonhomeomorphic Spaces 7210

7 Homeomorphism Problem and Topological Properties 72108 Information: Nonhomeomorphic Spaces 72109 Embeddings 731010 Equivalence of Embeddings 731011 Information 74

Chapter 2. Topological Properties 75

11. Connectedness 75111 Definitions of Connectedness and First Examples 75

112 Connected Sets 75

113 Properties of Connected Sets 76114 Connected Components 76115 Totally Disconnected Spaces 77116 Frontier and Connectedness 77117 Connectedness and Continuous Maps 77118 Connectedness on Line 78119 Intermediate Value Theorem and Its Generalizations 791110 Dividing Pancakes 791111 Induction on Connectedness 791112 Applications to Homeomorphism Problem 80

12. Path-Connectedness 81121 Paths 81122 Path-Connected Spaces 81123 Path-Connected Sets 82124 Properties of Path-Connected Sets 82125 Path-Connected Components 82126 Path-Connectedness Versus Connectedness 83127 Polygon-Connectedness 83128 Connectedness of Some Sets of Matrices 84

13. Separation Axioms 85

131 The Hausdorff Axiom 85

132 Limits of Sequence 85


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CONTENTS xi

133 Coincidence Set and Fixed Point Set 86

134 Hereditary Properties 86

135 The First Separation Axiom 86136 The Kolmogorov Axiom 87137 The Third Separation Axiom 88138 The Fourth Separation Axiom 88139 Niemytskis Space 891310 Urysohn Lemma and Tietze Theorem 89

14. Countability Axioms 90141 Set-Theoretic Digression. Countability 90142 Second Countability and Separability 90143 Embedding and Metrization Theorems 9114

4 Bases at a Point 91145 First Countability 92146 Sequential Approach to Topology 92147 Sequential Continuity 93

15. Compactness 94151 Definition of Compactness 94152 Terminology Remarks 94153 Compactness in Terms of Closed Sets 95154 Compact Sets 95155 Compact Sets Versus Closed Sets 95

156 Compactness and Separation Axioms 96

157 Compactness in Euclidean Space 96158 Compactness and Continuous Maps 97159 Closed Maps 971510 Norms inRn 98

16. Local Compactness and Paracompactness 99161 Local Compactness 99162 One-Point Compactification 99163 Proper Maps 100164 Locally Finite Collections of Subsets 100

165 Paracompact Spaces 101

166 Paracompactness and Separation Axioms 101167 Partitions of Unity 101168 Application: Making Embeddings from Pieces 101

17. Sequential Compactness 103171 Sequential Compactness Versus Compactness 103172 In Metric Space 103173 Completeness and Compactness 104174 Non-Compact Balls in Infinite Dimension 104175 p-Adic Numbers 104176 Induction on Compactness 10517

7 Spaces of Convex Figures 105


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CONTENTS xii

Problems for Tests 107

Chapter 3. Topological Constructions 109

18. Multiplication 109181 Set-Theoretic Digression. Product of Sets 109182 Product of Topologies 110183 Topological Properties of Projections and Fibers 110184 Cartesian Products of Maps 111185 Properties of Diagonal and Other Graphs 111186 Topological Properties of Products 112187 Representation of Special Spaces as Products 113

19. Quotient Spaces 114191 Set-Theoretic Digression. Partitions and EquivalenceRelations 114

192 Quotient Topology 115193 Topological Properties of Quotient Spaces 115194 Set-Theoretic Digression. Quotients and Maps 116195 Continuity of Quotient Maps 116196 Closed Partitions 117197 Open Partitions 117198 Set-Theoretic Digression:

Splitting a transitive relation

into equivalence and partial order 117199 Finite Topological Spaces 1181910 Simplicial schemes 1191911 Baricentric Subdivision of a Poset 119

20. Zoo of Quotient Spaces 121201 Tool for Identifying a Quotient Space with a Known

Space 121202 Tools for Describing Partitions 121203 Entrance to the Zoo 122204 Transitivity of Factorization 12320

5 Mobius Strip 124206 Contracting Subsets 124207 Further Examples 125208 Klein Bottle 125209 Projective Plane 1262010 You May Have Been Provoked to Perform an Illegal

Operation 1262011 Set-Theoretic Digression. Sums of Sets 1262012 Sums of Spaces 1262013 Attaching Space 127

2014 Basic Surfaces 128

21. Projective Spaces 130


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CONTENTS xiii

211 Real Projective Space of Dimension n 130

212 Complex Projective Space of Dimensionn 131

213 Quaternionic Projective Spaces 13122. Spaces of Continuous Maps 134

221 Sets of Continuous Mappings 134222 Topologies on Set of Continuous Mappings 134223 Topological Properties of Mapping Spaces 135224 Metric Case 135225 Interactions With Other Constructions 136226 Mappings X Y Zand X C(Y, Z) 136

Chapter 4. A Touch of Topological Algebra 138

23. Digression. Generalities on Groups 139231 The Notion of Group 139232 Additive and Multiplicative Notations 140233 Homomorphisms 141234 Subgroups 141

24. Topological Groups 143241 The Notion of Topological Group 143242 Examples of Topological Groups 143243 Self-Homeomorphisms Making a Topological Group

Homogeneous 143

244 Neighborhoods 144

245 Separaion Axioms 145246 Countability Axioms 145247 Subgroups 145248 Normal Subgroups 146249 Homomorphisms 1472410 Local Isomorphisms 1472411 Direct Products 148

25. Actions of Topological Groups 150251 Actions of Group in Set 150

252 Continuous Actions 150

25

3 Orbit Spaces 150254 Homogeneous Spaces 150

Part 2. Algebraic Topology 151

Chapter 5. Fundamental Group and Covering Spaces 153

26. Homotopy 153261 Continuous Deformations of Maps 153262 Homotopy as Map and Family of Maps 153263 Homotopy as Relation 154

264 Straight-Line Homotopy 154

265 Maps to Star Convex Sets 154


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CONTENTS xiv

266 Maps of Convex Sets 155

267 Easy Homotopies 155

268 Two Natural Properties of Homotopies 155269 Stationary Homotopy 1562610 Homotopies and Paths 1562611 Homotopy of Paths 156

27. Homotopy Properties of Path Multiplication 158271 Multiplication of Homotopy Classes of Paths 158272 Associativity 158273 Unit 159274 Inverse 159

28. Fundamental Group 161

281 Definition of Fundamental Group 161282 Why Index 1? 161283 High Homotopy Groups 161284 Circular loops 162285 The Very First Calculations 163286 Fundamental Group of Product 163287 Simply-Connectedness 164288 Fundamental Group of a Topological Group 165

29. The Role of Base Point 166291 Overview of the Role of Base Point 16629

2 Definition of Translation Maps 166293 Properties ofTs 166294 Role of Path 167295 High Homotopy Groups 167296 In Topological Group 168

Chapter 6. Covering Spaces and Calculation of Fundamental

Groups 169

30. Covering Spaces 169301 Definition of Covering 169

302 More Examples 169

303 Local homeomorphisms versus coverings 170304 Number of Sheets 171305 Universal Coverings 171

31. Theorems on Path Lifting 172311 Lifting 172312 Path Lifting 172313 Homotopy Lifting 173314 High-Dimensional Homotopy Groups of Covering Space173

32. Calculations of Fundamental Groups Using UniversalCoverings 174

321 Fundamental Group of Circle 174


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CONTENTS xv

322 Fundamental Group of Pro jective Space 175

323 Fundamental Groups of Bouquet of Circles 175

324 Algebraic Digression. Free Groups 175325 Universal Covering for Bouquet of Circles 177326 Fundamental groups of some finite

topological spaces 178

Chapter 7. Fundamental Group and Mappings 179

33. Induced Homomorphismsand Their First Applications 179

331 Homomorphisms Induced by a Continuous Map 179332 Fundamental Theorem of Algebra 18033

3 Generalization of Intermediate Value Theorem 181334 Winding Number 182335 Borsuk-Ulam Theorem 182

34. Retractions and Fixed Points 184341 Retractions and Retracts 184342 Fundamental Group and Retractions 184343 Fixed-Point Property. 185

35. Homotopy Equivalences 187351 Homotopy Equivalence as Map 187352 Homotopy Equivalence as Relation 187

353 Deformation Retraction 187

354 Examples 188355 Deformation Retraction Versus Homotopy Equivalence188356 Contractible Spaces 189357 Fundamental Group and Homotopy Equivalences 189

36. Covering Spaces via Fundamental Groups 191361 Homomorphisms Induced by Covering Projections 191362 Number of Sheets 191363 Hierarchy of Coverings 192364 Existence of subordinations 193

365 Micro Simply Connected Spaces 193

36

6 Existence of Coverings 194367 Automorphisms of Covering 194368 Regular Coverings 195369 Lifting and Covering Maps 196

Chapter 8. Cellular Techniques 197

37. Cellular Spaces 197371 Definition of Cellular Spaces 197372 First Examples 199373 Further Two-Dimensional Examples 200

374 Simplicial spaces 201

375 Topological Properties of Cellular Spaces 201


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CONTENTS xvi

376 Embedding to Euclidean Space 202

377 Euler Characteristic 203

378 Collaps 203379 Generalized collaps 204

38. One-Dimensional Cellular Spaces 206381 Homotopy Classification 206382 Dividing Cells 206383 Trees and Forests 207384 Simple Paths 207385 Maximal Trees 208

39. Fundamental Group of a Cellular Space 209

391 One-Dimensional Cellular Spaces 209

392 Generators 209393 Relators 209394 Writing Down Generators and Relators 210395 Fundamental Groups of Basic Surfaces 211396 Seifert - van Kampen Theorem 212

40. One-Dimensional Homology and Cohomology 213401 Why and What for 213402 One-Dimensional Integer Homology 213403 Zero Homologous Loops and Disks with Handles 214404 Description ofH1(X) in Terms of Free Circular Loops 21440

5 Homology and Continuous Maps 215406 One-Dimensional Cohomology 216407 Cohomology and Classification of Regular Coverings 216408 Integer Cohomology and Maps toS1 216409 One-Dimensional Homology Modulo 2 217

Part 3. Manifolds 219

Chapter 9. Bare Manifolds 221

41. Locally Euclidean Spaces 221

411 Definition of Locally Euclidean Space 221

412 Dimension 221413 Interior and Boundary 222

42. Manifolds 225421 Definition of Manifold 225422 Components of Manifold 225423 Making New Manifolds out of Old Ones 225424 Double 226425 Collars and Bites 226

43. Isotopy 228

431 Isotopy of Homeomorphisms 228

432 Isotopy of Embeddings and Sets 228


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CONTENTS xvii

433 Isotopies and Attaching 229

434 Connected Sums 230

44. One-Dimensional Manifolds 231441 Zero-Dimensional Manifolds 231442 Reduction to Connected Manifolds 231443 Examples 231444 Statements of Main Theorems 231445 Lemma on 1-Manifold Covered with Two Lines 232446 Without Boundary 233447 With Boundary 233448 Consequences of Classification 233449 Mapping Class Groups 233

45. Two-Dimensional Manifolds: General Picture 234451 Examples 234452 Ends and Odds 234453 Closed Surfaces 235

46. Triangulations 237461 Triangulations of Surfaces 237462 Triangulation as cellular decomposition 237463 Two Properties of Triangulations of Surfaces 237464 Scheme of Triangulation 238465 Examples 23846

6 Subdivision of a Triangulation 239467 Homotopy Type of Compact Surface with Non-EmptyBoundary 241

468 Triangulations in dimension one 241469 Triangualtions in higher dimensions 242

47. Handle Decomposition 243471 Handles and Their Anatomy 243472 Handle Decomposition of Manifold 243473 Handle Decomposition and Triangulation 244474 Regular Neighborhoods 245

475 Cutting 2-Manifold Along a Curve 245

48. Orientations 248481 Orientations of Edges and Triangles 248482 Orientation of Triangulation 248

49. Classical Approach to Topological Classificationof Compact surfaces 249

491 Families of Polygons 249492 Operations on Family of Polygons 250493 Topological and Homotopy Classification of Closed

Surfaces 250

494 Recognizing Closed Surfaces 251

495 Orientations 252


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CONTENTS xviii

496 More About Recognizing Closed Surfaces 252

497 Compact Surfaces with Boundary 253

498 Simply Connected Surfaces 25350. One-Dimensional mod2-Homology of Surfaces 254

501 Polygonal Paths on Surface 254502 Subdivisions of Triangulation 254503 Bringing Loops to General Position 255504 Cutting Surface Along Curve 256505 Curves on Surfaces and Two-Fold Coverings 257506 One-Dimensional Z2-Cohomology of Surface 257507 One-Dimensional Z2-Homology of Surface 258508 Poincare Duality 25850

9 One-Sided and Two-Sided Simple Closed Curves onSurfaces 258

5010 Orientation Covering and First Stiefel-Whitney Class 2585011 Relative Homology 258

51. Surfaces Beyond Classification 259511 Genus of Surface 259512 Systems of disjoint curves on a surface 259513 Polygonal Jordan and Schonflies Theorems 259514 Polygonal Annulus Theorem 259515 Dehn Twists 259

516 Coverings of Surfaces 259

517 Branched Coverings 259518 Mapping Class Group of Torus 259519 Braid Groups 259

52. Three-Dimensional Manifolds 260521 Poincare Conjecture 260522 Lens Spaces 260523 Seifert Manifolds 260524 Fibrations over Circle 260525 Heegaard Splitting and Diagrams 260Proofs and Comments 261


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Part 1

General Topology


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The goal of this part of the book is to teach the language of math-ematics. More specifically, one of its most important components: thelanguage of set-theoretic topology, which treats the basic notions relatedto continuity. The termgeneral topologymeans: this is the topology thatis needed and used by most mathematicians.

As a research field, it was completed a long time ago. A permanentusage in the capacity of a common mathematical language has polishedits system of definitions and theorems. Nowadays studying general topol-ogy really resembles studying a language rather than mathematics: oneneeds to learn a lot of new words, while proofs of all theorems are ex-tremely simple. On the other hand, the theorems are numerous, for theyplay the role of rules regulating usage of words.

We have to warn students, for whom this is one of the first mathe-matical subjects. Do not hurry to fall in love with it too seriously, donot let an imprinting happen. This field may seam to be charming, butit is not very active. It hardly provides as much room for exciting newresearch as most of other fields.


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CHAPTER 1

Generalities

1 Digression on SetsWe begin with a digression, which we would like to consider unnec-

essary. Its subject is the first basic notions of the naive set theory. Thisis a part of the common mathematical language, too, but even moreprofound than general topology. We would not be able to say anythingabout topology without this part (see the next section to make sure thatthis is not an exaggeration). Naturally, one may expect that naive settheory becomes familiar to a student when she or he studies Calculusor Algebra, the subjects which usually precede topology. If this is whatreally happened to you, please, glance through this section and move tothe next one.

11 Sets and ElementsIn any intellectual activity, one of the most profound action is gath-

ering objects into groups. The gathering is performed in minds and isnot accompanied with any action in the physical world. As soon as thegroup has been created and assigned with a name, it may be subject ofthoughts and arguments and, in particular, may be included into othergroups. In Mathematics there is an elaborated system of notions whichorganizes and regulate creating of those groups and manipulating them.This system is called the naive set theory, a slightly misleading name,because this is rather a language, than a theory.

The first words in this language are set and element. y a set weunderstand an arbitrary collection of various objects. An object includedinto the collection is called an element of the set. A set consists of itselements. It is formed by them. To diversify wording, the word set isreplaced by the word collection. Sometimes other words, such as class,family and group, are used in the same sense, but it is not quite safe,since each of these words is associated in the modern mathematics witha more special meaning, and hence should be used instead of the wordset cautiously.

Ifxis an element of a set A, we write x Aand say x belongs toAand Acontainsx. The sign

is a version of Greek letter epsilon, which

is the first letter of the Latin word element. To make formulas more

3


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1. DIGRESSION ON SETS 4

flexible, the formula x A is allowed to be written also as A x. Sothe origin of notation is ignored, but a more meaningful similarity to theinequality symbols < and > is emphasized. To state that x is not anelement ofA, we write x A or A x.

12 Equality of SetsA set is defined by its elements. It is nothing but a collection of its

elements. This manifests most sharply in the following principle: twosets are considered equal, if and only if they have the same elements. Inthis sense the wordsethas slightly disparaging meaning. When one callssomething a set, this shows, maybe unintentionally, a lack of interest to

whatever organization of the elements of this set.For example, when we say that a line is a set of points, we indicate

that two lines coincide if and only if they consist of the same points. Onthe other hand, we commit ourselves to consider all the relations betweenpoints on a line (e.g. the distance between points, the order of points onthe line) separately from the notion of line.

We may think of sets as boxes, which can be built effortlessly aroundelements, just to distinguish them from the rest of the world. The cost ofthis lightness is that such a box is not more than the collection of elementsplaced inside. It is a little more than just a name: it is a declaration of

our wish to think about this collection of things as of entity and not togo into details about the nature of its members-elements. Elements, inturn, may also be sets, but as long as we consider them elements, theyplay the role of atoms with their own original nature ignored.

In modern Mathematics the wordssetand elementare very commonand appear in most of texts. They are even overused. There are instanceswhen it is not appropriate to use them. For example, it is not goodto use the word element as a replacement for other, more meaningfulwords. When you call something an element, the set, whose elementis this one, should be clear. The word element makes sense only in a

combination with the word set, unless we deal with non-mathematicalterm (like chemical element), or a rare old-fashioned exception from thecommon mathematical terminology (sometimes the expression under thesign of integral is called aninfinitesimal element,in old texts lines, planesand other geometric images are called elements). Euclids famous bookon Geometry is called Elements.

13 The Empty SetThus, an element may not be without a set. However a set may

be without elements. There is a set which has no element. This set is

unique, because a set is defined completely by its elements. It is called


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the empty setand denoted by . Other notations, like , also were inuse, but has become common.

14 Basic Sets of NumbersBesides , there are few other sets, which are so important that have

their own unique names and notation. The set of all natural numbers,i.e., 1, 2, 3, 4, 5, ..., etc., is denoted by N. The set of all integernumbers, both positive (that is natural numbers) and negative and thezero, is denoted by Z. The set of all the rational numbers (add to theintegers those numbers which can be presented by fractions, like 2

3,7

5)

is denoted by Q. The set of all the real numbers (obtained by adjoiningto rational numbers the numbers like

2 and = 3.14 . . . ) is denoted

byR. The set of complex numbers is denoted by C.

15 Describing a Set by Listing of Its ElementsThe set presented by the list a, b, . . . , x of its elements is denoted

by symbol{a , b , . . . , x}. In other words, the list of objects enclosed in acurly brackets denotes the set, whose elements are listed. For example,{1, 2, 123} denotes the set which consists of numbers 1, 2 and 123. No-tation{a,x,A}means the set which consists of three elements, a,xandA, whatever these three letters denote.

1.1. What is{}? How many elements does it contain?1.2. Which of the following formulas are correct:

1) {, {}}; 2) {} { {}}; 3) {{}}?

A set consisting of a single element is called a singleton. This is anyset which can be presented as{a} for some a.

1.3. Is{{}}a singleton?

Notice that sets{1, 2, 3} and{3, 2, 1, 2} are equal, since they consistsof the same elements. At first glance, a list with repetition of elements

is never needed. There arises even a temptation to prohibit usage oflists with repetitions in such a notation. However, as it often happensto temptations to prohibit something, this would not be wise. In fact,quite often one cannot say a priori if there are repetitions or not. Forexample, the elements of the list may depend on parameter, and undercertain values of the parameters some entries of the list coincide, whilefor other values, they dont.

1.4. How many elements do the following sets contain?

1) {1, 2, 1}; 2) {1, 2, {1, 2}}; 3) {{2}};4)

{{1

}, 1

}; 5)

{1,

}; 6)

{{

},

};

7) {{}, {}}; 8) {x, 3x 1} for x R.


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16 Subsets

IfA andB are sets and every element ofA belongs also toB , we saythatA is a subset ofB, or B includes A, and write A B or B A.

The inclusion signs and recall the inequality signs < and> for a good reason: in the world of sets the inclusion signs are obviouscounterparts for the signs of inequalities.

1.A. Let a setA consists ofa elements, and a setB ofb elements. Provethat ifA B then a b.

Thus, the inclusion signs are not completely true counterparts of theinequality signs < and >. They are closer to

and

.

17 Properties of Inclusion1.B Reflexivity of Inclusion. Any set includes itself: A A holdstrue for any A.

Notice that there is no number asatisfying inequality a < a.

1.C The Empty Set Is Everywhere. Afor any set A. In otherwords, the empty set is present in each set as a subset.

Thus, each set A has two obvious subsets: the empty set and A

itself. A subset ofA different from and Ais called a proper subset ofA. This word is used when one does not want to consider the obvioussubsets (which are called improper).

1.D Transitivity of Inclusion. IfA, B and C are sets, AB andB C, then A C.18 To Prove Equality of Sets, Prove Inclusions

Working with sets, we need from time to time to prove that two sets,sayA and B , which may have emerged in quite different ways, are equal.The most common way to do this is provided by the following theorem.

1.E Criterium of Equality for Sets.A= B , if and only ifA B andB A.

19 Inclusion Versus Belonging1.F. x A, if and only if{x} A.

Despite this obvious relation between the notions of belonging andinclusion and similarity of the symbols and , the concepts arevery different. Indeed,AB means that Ais one of the elements ofB(that is one of indivisible pieces comprising B), whileA

B means that

A is made of some of the elements ofB.


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In particular, A A, while A A for any reasonable A. Thus,belonging is not reflexive. Yet another difference: belonging is not tran-sitive, while inclusion is.

1.G Non-Reflexivity of Belonging. Construct sets A and B suchthatA A, while B B. Cf. 1.B.1.H Non-Transitivity of Belonging. Construct sets A, B and Csuch that A B and B C, but A C. Cf. 1.D.

110 Defining a Set by a ConditionAs we know (see Section15), a set can be described by presenting

a list of its elements. This simplest way may be not available or, at least,be not the easiest one. For example, it is easy to say: the set of all thesolutions of the following equation and write down the equation. This isa reasonable description of the set. At least, it is unambiguous. Havingaccepted it, we may start speaking on the set, studying its properties,and eventually may be lucky to solve the equation and get the list of itssolutions. However the latter may be difficult and should not prevent usfrom discussing the set.

Thus we see another way for description of a set: to formulate theproperties which distinguish the elements of the set among elements of

some wider and already known set. Here is the corresponding notation:the subset of a set A consisting of elements x which satisfy conditionP(x) is denoted by{x A | P(x)}.

1.5. Present the following sets by lists of their elements (i.e., in the form{a , b , . . . })

(a) {x N | x


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Here the conjunction or should be understood in the inclusive way: thestatement x

Aor x

B means thatx belongs to at least oneof the

sets Aand B, but, maybe, to both of them.

A B A B A B

A B A BFigure 1. Disks A and B, their intersection A B andunion A B.

1.I Commutativity of and. For any sets Aand BA B =B A A B =B A.

1.6. Prove that for any setA

A A= A, A A= A, A =A and A = .1.7. Prove that for any sets A andB

A B, iff A B= A, iff A B= B.1.J Associativity of and. For any sets A, B and C

(A B) C=A (B C) and (A B) C=A (B C).

Associativity allows us do not care about brackets and sometimes

even omit them. One defines A B C= (A B) C= A (B C)and A B C= (A B) C=A (B C). However, intersection andunion of arbitrarily large (in particular, infinite) collection of sets can bedefined directly, without reference to intersection or union of two sets.Indeed, let be a collection of sets. The intersectionof the sets belongingto is the set formed by elements which belong to everyset, belongingto . This set is denoted byAA or

A A. Similarly, the union of

the sets belonging to is the set formed by elements which belong to atleast one of the sets belonging to . This set is denoted byAA or

A A.

1.K. The notions of intersection and union of arbitrary collection ofsets generalize the notions of intersection and union of two sets: for = {A, B}

C

C=A B andC

C=A B.

1.8. Enigma. How are related to each other the notions of system of equa-tions and intersection of sets?

1.L Two Distributivities. For any setsA, B andC

(A B) C= (A C) (B C).(1)(A B) C= (A C) (B C)(2)


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A A BB

C C C

(A B) C (A C) (B C)=

=

Figure 2. The left-hand side (A B) Cof the equality(1) and the sets A C B C, whose intersection is theright-hand side of the equation (1). A B.

In Figure 2 the first of two equalities of Theorem 1.L is illustrated bya sort of comics. Such comics are calledVenn diagrams. They are very

useful and we strongly recommend to draw them for each formula aboutsets.

1.M. Draw a Venn diagram illustrating (2). Prove (1) and (2) tracingall the details of the proofs in Venn diagrams. Draw Venn diagramsillustrating all formulas below in this section.

1.9. Enigma. Generalize Theorem1.L to the case of arbitrary collection ofsets.

1.N Yet Another Pair of Distributivities. LetA be a set and bea set consisting of sets. Then

A B

B= B

(A B) and A B

B= B

(A B).

112 Different DifferencesAdifference A B of sets Aand B is the set of those elements ofA

which do not belong to B . Here it is not assumed that A B.IfA B, the set A B is called also the complementofB in A.

1.10. Prove that for any setsA and B their unionA B can be representedas the union of the following three sets: A B, B A and A

B, and that

these sets are pairwise disjoint.

1.11. Prove thatA (A B) = A B for any setsA and B .1.12. Prove thatA B, if and only ifA B= .1.13. Prove that A (B C) = (A B) (A C) for any sets A,B andC.

The set (A B) (B A) is called the symmetric differenceof setsA and B. It is denoted by A B.

1.14. Prove that for any sets A and B

A B= (A B) (A B)


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A B A B A B

B A A B A B

Figure 3. Differences of disks Aand B.

1.15 Associativity of Symmetric Difference. Prove that for any setsA, B andC

(A B) C=A (B C).

1.16. Enigma. Find a symmetric definition of symmetric difference (A B) Cof three sets and generalize it to any finite collection of sets.

1.17 Distributivity. Prove that (A B) C= (A C) (B C) for anysetsA, B and C.1.18. Does the following equality hold true for any sets A, B C

(A B) C= (A C) (B C)?

Proofs and Comments

1.A The question is so elementary that it is difficult to find moreelementary facts, which a proof can be based on. What does it mean thatAconsists ofa elements? It means, say, that we can count elements ofA

one by one assigning to them numbers 1, 2, 3, and the last element willget number a. It is known that the result does not depend on the orderin which we count. (In fact, one can develop the set theory, which wouldinclude a theory of counting, and in which this is a theorem. But sincewe have no doubts in this fact, let us use it without proof.) Thereforewe can start counting of elements ofB with counting the elements ofA.The counting of elements ofA will be done, first, and then, if there aresome elements ofB which are not in A, counting will continue. Thus thenumber of elements in A is less than or equal to the number of elementsin B.

1.B Recall that, by the definition of inclusion, A

B means thateach element of A is an element of B. Therefore the statement thatwe have to prove can be rephrased as follows: each element ofA is anelement ofA. This is a tautology.

1.C Recall that, by the definition of inclusion, AB means thateach element ofA is an element ofB. Thus we need to prove that anyelement ofbelongs to A. This is correct, because there is no elementsin . If you are not satisfied with this argument (since it sounds toocrazy), let us resort to a question, whether this can be wrong. How canit happen that is not a subset ofA? It could happen, only if there wasan element of which would not be an element ofA. But there is no

such an element in , because has no elements at all.


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1.D We have to prove that each element ofA is an element ofC.Let x

A. Since A

B , it follows that x

B . Since b

C, the latter

(i.e.,x B) implies x C. This is what we had to prove.1.E We have already seen thatA A. Hence ifA = B thenA B

and B A. On the other hand, A B means that each element ofA belongs to B and B A means that each element of B belongs toA. Hence A and B have the same elements, which means that they areequal.

1.G It is easy to construct a set A with A A. Take A = , orA = N, or A ={1},...A set B such that B B is a strange creature.It would not appear in real problems, unless you think really globally.

Take for B the set of all sets. Mathematicians avoid such sets. Thereare good reasons for this. If we consider the set of all sets, why not toconsider the set Yof all the setsXsuch thatX X? DoesYbelongs toitself? IfY Y thenY Y, since each element XofYhas the propertythat X X. IfY Y then Y Y since Y is the set of ALL the setsXsuch that XX. This contradiction shows that our definition ofYdoes not make sense. An easy way to avoid this paradox is to prohibitconsideration of sets with the property X X. The the set of all sets isnot a legitimate set.

1.H TakeA= {1},B = {{1}} andC= {{{1}}}. It is more difficultto construct setsA,B andCsuch thatA

B,B

C, andA

C. Take

A= {1}, B = {{1}},C= {{1}, {{1}}}.

Hints, Comments, Advises, Solutions, and Answers

1.1 The set{}consists of one element, which is the empty set . Ofcourse, this element itself is the empty set and contains no element, but theset{} consists of a single element .

1.2 1) and 2) are correct, 3) is not.

1.3 Yes,{{}}is a singleton.1.4 2, 3, 1, 2, 2, 2, 1, 2 for x = 12 and 1 ifx = 12 .1.5 (a){1, 2, 3, 4}; (b){}; (c){1, 2, 3, 4, 5, 6, . . .}1.8 The set of solutions for a system of equations is equal to the in-

tersection of the sets of solutions of individual equations belonging to thesystem.


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2 Topology in a Set21 Definition of Topological Space

Let Xbe a set. Let be a collection of its subsets such that:(a) the union of a collection of sets, which are elements of , belongs to

;(b) the intersection of a finite collection of sets, which are elements of

, belongs to ;(c) the empty setand the whole Xbelong to .

Then is called a topological structureor just a topology1 inX;

the pair (X, ) is called a topological space;

an element ofXis called a pointof this topological space; an element of is called anopen setof the topological space (X, ).

The conditions in the definition above are called the axioms of topologicalstructure.

22 The Simplest ExamplesA discrete topological space is a set with the topological structure

which consists of all the subsets.

2.A. Check that this is a topological space, i.e., all axioms of topological

structure hold true.An indiscrete topological spaceis the opposite example, in which the

topological structure is the most meager. It consists only ofXand .

2.B. This is a topological structure, is it not?

Here are slightly less trivial examples.

2.1. Let X be the ray [0, +), and consists of , X, and all the rays(a, +) with a 0. Prove that is a topological structure.2.2. LetXbe a plane. Let consist of,X, and all open disks with centerat the origin. Is this a topological structure?

2.3. Let Xconsist of four elements: X ={a,b,c,d}. Which of the follow-ing collections of its subsets are topological structures in X, i.e., satisfy theaxioms of topological structure:(a) , X,{a},{b},{a, c},{a,b,c},{a, b};(b) , X,{a},{b},{a, b},{b, d};(c) , X,{a,c,d},{b,c,d}?

The space of2.1 is called anarrow. We denote the space of2.3(a) by .It is a sort of toy space made of 4 points. Both of these spaces, as well as the

the arrow:

space of2.2, are not important, but provide good simple examples.

1Thus is important: it is called by the same word as the whole branch ofmathematics. Of course, this does not mean that coincides with the subject of

topology, but everything in this subject is related to .

12


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2. TOPOLOGY IN A SET 13

23 The Most Important Example: Real Line

Let Xbe the set Rof all real numbers, be the set of unions of allintervals (a, b) with a, b R.2.C. Check if satisfies the axioms of topological structure.

This is the topological structure which is always meant when R isconsidered as a topological space (unless other topological structure isexplicitly specified). This space is called usually the real line and thestructure is referred to as the canonical or standardtopology in R.

24 Additional Examples

2.4. Let X be R, and consists of empty set and all the infinite subsets ofR. Is a topological structure?

2.5. Let X be R, and consists of empty set and complements of all finitesubsets ofR. Is a topological structure?

The space of2.5is denoted by RT1 and called the line with T1-topology.

2.6. Let (X, ) be a topological space and Ybe the set obtained from X byadding a single element a. Is

{{a} U : U } {}a topological structure inY?

2.7. Is the set{, {0}, {0, 1}} a topological structure in{0, 1}?

In Problem 2.6, if topology discrete, the topology in Y is called aparticular point topology or topology of everywhere dense point. The topologyin Problem2.7is a particular point topology; it is called also the topology ofconnected pair of points or Sierpinski topology.

2.8. List all the topological structures in a two-element set, say, in{0, 1}.

25 Using New Words: Points, Open and Closed Sets

Recall that, for a topological space (X, ), elements ofXare calledpoints, and elements of are called open sets.2

2.D. Reformulate the axioms of topological structure using the wordsopen set wherever possible.

A setF Xis said to beclosedin the space (X, ) if its complementX F is open (i.e., X F ).

2The letter stands for the letter O which is the initial of the words with thesame meaning: Open in English, Otkrytyj in Russian, Offen in German, Ouvert in

French.


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26 Set-Theoretic Digression. De Morgan Formulas

2.E. Let be an arbitrary collection of subsets of a set X. Then

(3) XA

A=A

(X A)

(4) XA

A=A

(X A).

Formula (4) is deduced from (3) in one step, is it not? These formulas arenonsymmetric cases of a single formulation, which contains in a symmetricway sets and their complements, unions and intersections.

2.9. Enigma. Find such a formulation.

27 Properties of Closed Sets2.F. Prove that:

(a) the intersection of any collection of closed sets is closed;(b) union of any finite number of closed sets is closed;(c) empty set and the whole space (i.e., the underlying set of the topo-

logical structure) are closed.

28 Being Open or ClosedNotice that the property of being closed is not a negation of the

property of being open.

(They are not exact antonyms in everyday usage, too).

2.G. Find examples of sets, which

(a) are both open, and closed simultaneously;(b) are neither open, nor closed.

2.10. Give an explicit description of closed sets in(a) a discrete space; (b) an indiscrete space;(c) the arrow; (d) ;(e) RT1 .

2.H. Is a closed segment [a, b] closed in R.

Concepts of closed and open sets are similar in a number of ways.The main difference is that the intersection of an infinite collection ofopen sets does not have to be necessarily open, while the intersection ofany collection of closed sets is closed. Along the same lines, the unionof an infinite collection of closed sets is not necessarily closed, while the

union of any collection of open sets is open.


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2.11. Prove that the half-open interval [0, 1) is neither open nor closed in R,but can be presented as either the union of closed sets or intersection of open

sets.

2.12. Prove that every open set of the real line is a union of disjoint openintervals.

2.13. Prove that the set A = {0}

1

n

n=1

is closed in R.

29 Cantor SetLetKbe the set of real numbers which can be presented as sums of series of the

form

k=1

ak3k

with ak = 0 or 2. In other words, Kis the set of real numbers which

in the positional system with base 3 are presented as 0.a1a2 . . . ak. . . without digit 1.

2:A. Find a geometric description ofK.

2:A.1. Prove that(a) Kis contained in [0, 1],(b) Kdoes not intersect

13 ,

23

,

(c) Kdoes not intersect3s+13k

, 3s+23k

for any integers k and s.

2:A.2. PresentKas [0, 1] with an infinite family of open intervals removed.

2:A.3. Try to draw K.

The set K is called the Cantor set. It has a lot of remarkable properties and is

involved in numerous problems below.

2:B. Prove that K is a closed set in the real line.

210 Characterization of Topology in Terms of Closed Sets2.14. Prove that if a collectionF of subsets of X satisfies the followingconditions:(a) the intersection of any family of sets fromFbelongs toF;(b) the union of any finite number sets fromFbelongs toF;(c) and Xbelong toF,

then

Fis the set of all closed sets of a topological space (which one?).

2.15. List all collections of subsets of a three-element set such that thereexist topologies, in which these collections are complete sets of closed sets.

211 Topology and Arithmetic Progressions2.16*. Consider the following property of a subsetFof the set N of naturalnumbers: there exists N N such that F does not contain an arithmeticprogression of length greater than N. Prove, that subsets with this propertytogether with the whole N form a collection of closed subsets in some topologyin N.

Solving this problem, you probably are not able to avoid the following

combinatorial theorem.


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2.17 Van der Waerdens Theorem*. For every n N there existsN

N such that for any A

{1, 2, . . . , N

}, either A or

{1, 2, . . . , N

} A

contains an arithmetic progression of length n.

212 NeighborhoodsBy a neighborhood of a point one means any open set containing

this point. Analysts and French mathematicians (following N. Bourbaki)prefer a wider notion of neighborhood: they use this word for any setcontaining a neighborhood in the sense above.

2.18. Give an explicit description of all neighborhoods of a point in

(a) a discrete space; (b) an indiscrete space;(c) the arrow; (d) ;(e) connected pair of points; (f) particular point topology.

Proofs and Comments

2.A What should we check? The first axiom reads here that theunion of any collection of subsets ofX is a subset ofX? Well, this isright. IfA Xfor each A thenAA X. Indeed, take arbitrarypointb

AA. Since it belongs to the union, it belongs to at least one

ofA , and since A X, it belongs to X. Exactly in the same wayone checks the second axiom. Finally, of course, Xand X X.

2.B Yes, it is. Here we can list all the collections of sets that weneed to consider. If one of the united sets is X then the union is X.What if it is not there? Then what is there? Empty set, at most. Thenthe union is also empty. With intersections the situation is simialr. Ifone of the sets to intersect is the the intersection is . If it is notthere, then what is? Only the whole X. Then the intersection equalsX.

2.C First, show that

AA

BB =

A,B(A B). Therefore if

A and B are intervals then the right-hand side is a union of intervals.

If you think that a set which is a union of intervals is too simple,please, try to answer the following question (which has nothing to do withthe problem under consideration, though). Let{rn}n=1 = Q (i. e., wenumbered all the rational numbers). Prove that

(r 2n; r + 2n) = R,

although this is a union of some intervals, which contains all (!) therational numbers.

2.D The union of any collection of open sets is open. The intersec-tion of any finite collection of open sets is open. The empty set and the

whole space are open.


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2.E(a)

x A(X A) A x X A x / A x / AA

x X AA

(b) Replace both sides of the formula by their complements in X andput B=X A.

2.G In any topological space the empty set and the whole space

are both open and closed. In a discrete space any set is both open andclosed. Semiopen interval is neither open nor closed on the line. Cf. alsothe next problem.

2.H Yes, it is, because R [a; b] = (; a) (b; +) is open.


2.1 The solution is based on the equality(a; +) = (infa; +).Prove it. By the way the collection of closed rays [a; +) is not a topologicalstructure, since it may happen that

[a; +

) = (a0; +

) (find an example).

2.2 Yes, it is. A proof coincides almost literally with the solution of thepreceding problem.

2.3 The main point here is to realize that the axioms of topologicalstructure are conditions on the collection of subsets and if these conditionsare satisfied then the collection is called a topological structure. The secondcollection is not a topological structure, because the sets{a},{b, d}are con-tained in it, while{a,b,d} ={a} {b, d} is not. Find two elements of thethird collection such that there intersection does not belong to it. By thisyou would prove that this is not a topology. Finally, it is not difficult to seethat all the unions and intersections of elements of the first collection stillbelong to the first collection.

2.10 The following sets are closed

(a) in a disctrete space: all sets;(b) in an indiscrete: only those which are also open, that is the empty set

and the whole space;(c) in the arrow: , the whole space and segments of the form [0; a];(d) in : setsX, , {b,c,d}, {a,c,d}, {b, d}, {d}, {c, d};(e) inRT1 : all finite sets and the whole R.

2.11 Here it is important to overcome the feeling that the question iscompletely obvious. Why is not (0, 1] open? If (0; 1] =(a; b) then 1(a0 ; b0) for some 0, hence b0 > 1, and it follows that(a; b)= (0; 1].Similarly

R (0; 1] = (; 0] (1;+)


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is not open. On the other hand,

(0; 1] =

n=1

1n ; 1=

n=10;n + 1n .

2.14 Check that = {U| X U F} is a topological structure.2.15 Control indication: there number of such collections is 14.

2.16 The conditions (a) and (c) from 2.14 are obviously satisfied. Toprove (b), let us use 2.17. Let sets A and B do not contain arithmeticprogression of length n. If the set AB contained a sufficiently longprogression, in one of the original sets there would be a progression of lengthn.

2.18 By this point you have to learn already everything needed for

solving this problem, and must solve it on your own. Please, dont be lazy.


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3 Bases31 Definition of Base

Usually the topological structure is presented by describing its part,which is sufficient to recover the whole structure. A collection of opensets is called a base for a topology, if each nonempty open set is a unionof sets belonging to . For instance, all intervals form a base for the realline.

3.1. Are there different topological structures with the same base?

3.2. Find some bases of topology of(a) a discrete space; (b) ;

(c) an indiscrete space; (d) the arrow.Try to choose the bases as small as possible.

3.3. Describe all topological structures having exactly one base.

3.4. Prove that any base of the canonical topology in R can be diminished.

32 When a Collection of Sets is a Base3.A. A collectionof open sets is a base for the topology, iff for any opensetUand any pointx Uthere is a setV such thatx V U.3.B. A collection of subsets of a setX is a base for some topology in

X, iffX is a union of sets of and intersection of any two sets of isa union of sets in.

3.C. Show that the second condition in 3.B(on intersection) is equiva-lent to the following: the intersection of any two sets of contains, to-gether with any of its points, some set of containing this point (cf. 3.A).

33 Bases for PlaneConsider the following three collections of subsets ofR2:

2 which consists of all possible open disks (i.e., disks without itsboundary circles);

which consists of all possible open squares (i.e., squares withouttheir sides and vertices) with sides parallel to the coordinate axis;

1 which consists of all possible open squares with sides parallel to thebisectors of the coordinate angles.

(Squares of and 1 are defined by inequalities max{|xa|,|yb|} < and|x a| + |y b| < , respectively.)3.5. Prove that every element of 2 is a union of elements of .

3.6. Prove that intersection of any two elements of 1 is a union of elementsof 1.

3.7. Prove that each of the collections 2, , 1 is a base for some topo-logical structure in R2, and that the structures defined by these collections

coincide.

19


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3. BASES 20

Figure 1. Elements of (left) and 1 (right).

34 SubbasesLet (X, ) be a topological space. A collection of its open subsets is

called a subbasefor , provided the collection

= {V|V = ki=1Wi, Wi , k N}of all finite intersections of sets belonging to is a base for .

3.8. Prove that for any set Xa collection of its subsets is a subbase of atopology in X, iff = and X= WW.

35 Infiniteness of the Set of Prime Numbers3.9. Prove that all infinite arithmetic progressions consisting of natural num-bers form a base for some topology in N.

3.10. Using this topology prove that the set of all prime numbers is infinite.

36 Hierarchy of TopologiesIf 1 and 2 are topological structures in a set Xsuch that 1 2

then 2 is said to be finerthan 1, and 1 coarserthan 2. For instance,among all topological structures in the same set the indiscrete topologyis the coarsest topology, and the discrete topology is the finest one, is itnot?

3.11. Show that T1-topology (see Section2) is coarser than the canonicaltopology in the real line.

3.12. Enigma. Let 1and 2be bases for topological structures 1and 2in a set X. Find necessary and sufficient condition for 1 2 in terms ofthe bases 1 and 2 without explicit referring to 1 and 2 (cf. 3.7).

Bases defining the same topological structure are said to beequivalent.

3.D. Enigma. Formulate a necessary and sufficient condition for twobases to be equivalent without explicit mentioning of topological struc-tures defined by the bases. (Cf. 3.7: bases 2, , and 1 must satisfy

the condition you are looking for.)


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3. BASES 21

Proofs and Comments

3.A Let be a base of and U . Present U as a union ofelements of . Each point x Uis contained in some of these sets. Sucha set can be chosen as V. It is contained in U, since it participates in aunion which is equal to U.

Vice versa, assume that for any U and any point x U thereexists a set V such that x V U, and show that is a base of. For this we need to prove that any U can be represented as aunion of elements of . For each point x Uchoose according to theassumption a set Vx such that x Vx U and considerxUVx.Notice that

xUVx

U, since Vx

U for each x

U. On the other

hand, each point x U is contained in its Vx and hence inxUVx.Therefore U xUVx. Thus, U= xUVx.

3.B Assume that is a base of a topology. ThenX, being an openset in any topology, can be presented as a union of some sets belonging to|GS. The intersection of any two sets belonging to is open, thereforeit also can be presented as a union of base sets.

Vice versa, assume that is a collection of subsets ofX such thatXis a union of sets belonging to and the intersection of any two setsbelonging to is a union of sets belonging to . Let us prove that the

set of unions of all the collections of elements of satisfies the axiomsof topological structure. The first axiom is obviously satisfied, since theunion of some unions is a union. Let us prove the second axiom (theintersection of two open sets is open). Let U =A V =B,A, B . Then U V = (A) (B) =,(A B), andsince, by the assumpiton,A Bcan be presented as union of elementsof , the intersection U V can be presented in this form, too. In thethird axiom, we need to check only the part concerning the whole X. Bythe assumption,Xis a union of sets belonging to .

3.D Let 1 and 2 be bases of topological structures 1and 2 in a

setX. Obviously, 1 2, iff U 1x U V 2 : x V U.Now recall that 1 = 2 1 2 and 2 1.


3.1 Of course, not! A topological structure is recovered from its baseas the set of unions of all collections of sets which belong to the base.

3.2

(a) A discrete space admits a base consisting of all one-point subsets of thespace and this base is minimal. (why?)

(b) For a base in one can take, say,{a},{b},{a, c},{a,b,c,d}.


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3. BASES 22

(c) In indiscrete space the minimal base is formed by a single set, the wholespace.

(d) In the arrow{[0, +), (r, +)}rQ+ is a base.3.3 The whole topological structure is its own base. So, the question is

when this is the only base. In such a space any open set cannot be representedas a union of two open sets distinct from it. Hence open sets are linearlyordered by inclusion. Moreover, the space should contain only finite numberof open sets, since otherwisean open set could be obtained as a union ofinfinite increasing sequence of open sets.

3.4 We will show that removing of any element from any base of thestandard topology of the line gives a base of the same topology! Let Ube anarbitrary element of a base. It can be presented as a union of open intervals,which are shorter than distance between some two points ofU. We would need

at least two such intervals. Each of the intervals, in turn, can be presented asa union of sets of the base under consideration. U is not involved into theseunions, since it is not contained in so short intervals. Hence U is a union ofelements of the base distinct from Uand it can be replaced by this union ina presentation of an open set as a union of elements of the base.

3.5, 3.6 In solution of each of these problems the following easy lemmamay help: A=

B, where B B,iff x A Bx B: x Bx A.

3.7 The statement: Bis a base of a topological structure is equivalentto the following: the set of unions of all collections of sets belonging to Bis atopological structure. 1 is a base of some topology by 3.Band 3.6. So, youneed to prove analogues of3.6for 2 and . To prove that the structuresdefined, say, by bases 1 and 2, you need to prove that a union of disks can

be presented as a union of squares and vice versa. Is it enough to prove thata disk is a union of squares? What is the simplest way to do this (cf. ouradvice concerning3.5and 3.6)?

3.9 Observe that intersection of arithmetic progressions is an arithmeticprogression.

3.10 Since the sets{i, i + d, i + 2d , . . .}, i = 1, . . . , d, are open, pairwisedisjoint and cover the whole N, it follows that each of them is closed. Inparticular, for each prime number p the set{p, 2p, 3p, . . .} is closed. Alltogether the sets of the form {p, 2p, 3p, . . .}cover N {1}. Hence if the set ofprime numbers was finite, the set {1} would be open. But it cannot presentedas union of arithmetic progressions.

3.11 Inclusion 1 2 means that a set open in the first topology(i.e., belonging to 1) belongs also to 2. Therefore, you should prove thatR {xi}ni=1 is open in the canonical topology of the line.

3.12 1 2, iff U 1 x U V 2: x V U.


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4 Metric Spaces41 Definition and First Examples

A function : X X R+ ={ x R| x 0 } is called a metric(or distance) in X, if(a) (x, y) = 0, iffx= y ;(b) (x, y) =(y, x) for every x, y X;(c) (x, y) (x, z) + (z, y) for every x, y, z X.

The pair (X, ), where is a metric in X, is called a metric space.The condition (c) is triangle inequality.

4.A. Prove that for any set X

: X X R+: (x, y) 0, ifx= y ;1, ifx =y

is a metric.

4.B. Prove that R R R+: (x, y) |x y| is a metric.4.C. Prove that Rn Rn R+: (x, y)

ni=1(xi yi)2 is a metric.

Metrics4.Band 4.Care always meant when R and Rn are consideredas metric spaces unless another metric is specified explicitly. Metric 4.Bis a special case of metric 4.C. These metrics are called Euclidean.

42 Further Examples4.1. Prove that Rn Rn R+: (x, y) maxi=1,...,n |xi yi| is a metric.4.2. Prove that Rn Rn R+: (x, y)

ni=1 |xi yi| is a metric.

Metrics in Rn introduced in4.C4.2are included in infinite series of themetrics

(p) : (x, y) n

i=1

|xi yi|p 1p

, p 1.

4.3. Prove that (p) is a metric for any p 1.

4.3.1 Holder Inequality. Prove thatn

i=1

xiyi

ni=1

xpi

1/p ni=1

yqi

1/qifxi, yi 0, p,q >0 and 1p + 1q = 1.

Metric of4.Cis (2), metric of4.2is (1), and metric of4.1can be denotedby() and adjoined to the series since

limp+

ni=1

api

1p

= max ai,

for any positivea1, a2, . . . , an.

23


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4. METRIC SPACES 24

4.4. Enigma. How is this related to 2, , and 1 from Section3?

For a real numberp 1 denote by l(p) the set of sequences x= {xi}i=1,2,...such that the series

i=1 |x|p converges.4.5. Prove that for any two elements x, y l(p) the seriesi=1 |xi yi|pconverges and that

(x, y)

i=1

|xi yi|p1p

, p 1

is a metric in l (p).

43 Balls and SpheresLet (X, ) be a metric space, leta be its point, and let r be a positive

real number. The sets

Br(a) = { x X| (a, x)< r },(5)Dr(a) = { x X| (a, x) r },(6)Sr(a) = { x X| (a, x) =r }(7)

are called, respectively, open ball, closed ball, and sphere of the space(X, ) with center at a and radius r.

44 Subspaces of a Metric Space

If (X, ) is a metric space and AX, then the restriction of metrictoA Ais a metric in A, and (A, AA) is a metric space. It is calleda subspace of (X, ).

The ball D1(0) and sphere S1(0) in Rn (with Euclidean metric, see

4.C) are denoted by symbols Dn and Sn1 and called n-dimensional balland (n1)-dimensional sphere. They are considered as metric spaces(with the metric restricted from Rn).

4.D. Check that D1 is the segment [1, 1]; D2 is a disk; S0 is the pairof points

{1, 1

};S1 is a circle; S2 is a sphere; D3 is a ball.

The last two statements clarify the origin of terms sphereandball(inthe context of metric spaces).

Some properties of balls and spheres in arbitrary metric space resem-ble familiar properties of planar disks and circles and spatial balls andspheres.

4.E. Prove that for points x and a of any metric space and any r >(a, x)

Dr(a,x)(x) Dr(a).4.6. Enigma. What ifr < (x, a)? What is an analogue for the statement

of Problem 4.Ein this case?


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4. METRIC SPACES 25

45 Surprising Balls

However in other metric spaces balls and spheres may have rather sur-prising properties.

4.7. What are balls and spheres in R2 with metrics of4.1 and 4.2(cf. 4.4)?

4.8. Find D1(a),D 12

(a), and S12

(a) in the space of4.A.

4.9. Find a metric space and two balls in it such that the ball with thesmaller radius contains the ball with the bigger one and does not coincidewith it.

4.10. What is the minimal number of points in the space which is requiredto be constructed in 4.9.

4.11. Prove that in 4.9 the big radius does not exceed double the smaller

radius.

46 Segments (What Is Between)4.12. Prove that the segment with end points a, b Rn can be described as

{ x Rn | (a, x) + (x, b) =(a, b) },where is the Euclidean metric.

4.13. How do the sets defined as in 4.12 look like with of 4.1 and 4.2?(Consider the case n = 2 if it appears to be easier.)

4

7 Bounded Sets and BallsA subsetA of a metric space (X, ) is said to be bounded, if there is a

numberd >0 such that (x, y)< dfor anyx, y A. The greatest lowerbound of such d is called the diameter ofAand denoted by diam(A).

4.F. Prove that a set A is bounded, iff it is contained in a ball.

4.14. What is the relation between the minimal radius of such a ball anddiam(A)?

48 Norms and Normed SpacesLetXbe a vector space (over R). Function X R +: x ||x|| is called

a norm if(a) ||x||= 0, iffx = 0;(b) ||x||= ||||x|| for any R andx X;(c) ||x + y|| ||x|| + ||y|| for anyx, y X.

4.15. Prove that ifx ||x|| is a norm then: X X R+: (x, y) ||x y||

is a metric.

The vector space equipped with a norm is called a normed space. Themetric defined by the norm as in 4.15turns the normed space into the metric

one in a canonical way.


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4. METRIC SPACES 26

4.16. Look through the problems of this section and figure out which of themetric spaces involved are, in fact, normed vector spaces.

4.17. Prove that every ball in the normed space is a convex3 set symmetricwith respect to the center of the ball.

4.18*. Prove that every convex closed bounded set in Rn, which is symmet-ric with respect to its center and is not contained in any affine space exceptRn itself, is the unit ball with respect to some norm, and that this norm isuniquely defined by this ball.

49 Metric Topology4.G. The collection of all open balls in the metric space is a base forsome topology (cf. 3.A, 3.B and 4.E).

4.G.1 Lemma. In any metric space, Br(a) Br(a,x)(x) for any point a,real number r >0 and point x Br(a).

This topology is calledmetric topology. It is said to be inducedby themetric. This topological structure is always meant whenever the metricspace is considered as a topological one (for instance, when one saysabout open and closed sets, neighborhoods, etc. in this space).

4.H. Prove that the standard topological structure in R introduced inSection2 is induced by metric (x, y) |x y|.

4.19. What topological structure is induced by the metric of4.A?

4.I. A set is open in a metric space, iff it contains together with any itspoint a ball with center at this point.

410 Openness and Closedness of Balls and Spheres

4.20. Prove that a closed ball is closed (with respect to the metric topology).

4.21. Find a closed ball, which is open (with respect to the metric topology).

4.22. Find an open ball, which is closed (with respect to the metric topology).

4.23. Prove that a sphere is closed.

4.24. Find a sphere, which is open.

3Recall that a setAis said to beconvexif for anyx, y Athe segment connectingx, y is contained in A. Of course, this definition is based on the notion of segment,so it makes sense only for subsets of spaces, where the notion of segment connecting

two point is defined. This is the case in vector and affine spaces over R


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4. METRIC SPACES 27

411 Metrizable Topological Spaces

A topological space is said to be metrizableif its topological structureis induced by some metric.

4.J. An indiscrete space is not metrizable unless it consists of a singlepoint (it has too few open sets).

4.K. A finite space is metrizable iff it is discrete.

4.25. Which topological spaces described in Section2 are metrizable?

412 Equivalent Metrics

Two metrics in the same set are said to be equivalent if they inducethe same topology.

4.26. Are the metrics of4.C, 4.1, and 4.2 equivalent?

4.27. Prove that metrics 1, 2 in X are equivalent if there are numbersc, C >0 such that

c1(x, y) 2(x, y) C1(x, y)for any x, y X.4.28. Generally speaking the inverse is not true.

4.29. Enigma. Hence the condition of the equivalence of metrics formulatedin 4.27can be weakened. How?

4.30. Metrics (p) in Rn defined right above Problem 4.3are equivalent.

4.31*. Prove that the following two metrics1, C in the set of all contin-uous functions [0, 1] Rare not equivalent:

1(f, g) =

10

f(x) g(x)dx; C(f, g) = maxx[0,1]

f(x) g(x).Is it true that topological structure defined by one of them is finer thananother?

4

13 Ultrametric

A metric is called an ultrametricif it satisfies to ultrametric triangle inequality:

(x, y) max{(x, z), (z, y)}for any x, y, z .

A metric space (X, ) with ultrametric is called an ultrametric space.

4:A. Check that only one metric in 4.A4.2is ultrametric. Which one?

4:B. Prove that in an ultrametric space all triangles are isosceles (i.e., for any threepointsa, b, c two of the three distances (a, b),(b, c),(a, c) are equal).

4:C. Prove that in a ultrametric space spheres are not only closed (cf. 4.23) but also

open.


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4. METRIC SPACES 28

The most important example of ultrametric is p-adic metric in the set Q of allrational numbers. Let p be a prime number. For x, y

Q, present the difference

x y as rsp, wherer,s, and are integers, andr,s are relatively prime with p. Put(x, y) =p.

4:D. Prove that this is an ultrametric.

414 Operations with Metrics4.32. Prove that if1, 2 are metrics in X then1+ 2 and max{1, 2}arealso metrics. Are the functions min{1, 2}, 1

2, and12 metrics?

4.33. Prove that if : X X R+ is a metric then(a) function

(x, y) (x, y)1 + (x, y)

is a metric;(b) function

(x, y) f(x, y)is a metric, iff satisfies the following conditions:

(1) f(0) = 0,(2) fis a monotone increasing function, and(3) f(x + y) f(x) + f(y) for any x, y R.

4.34. Prove that metrics and

1 + are equivalent.

415 Distance Between Point and SetLet (X, ) be a metric space,A X,b X. The inf{ (b, a) | a A }

is called a distance from the pointbto the setAand denoted by (b, A).

4.L. Let Abe a closed set. Prove that (b, A) = 0, iffb A.

4.35. Prove that|(x, A) (y, A)| (x, y) for any set A and points x, yof the same metric space.

416 Distance Between SetsLetA and B be bounded subsets in the metric space (X, ). Put

d(A, B) = max

supaA

(a, B), supbB

(b, A)

.

This number is called the Hausdorff distance betweenA and B .

4:E. Prove that the Hausdorff distance in the set of all bounded subsets of a metricspace satisfies the conditions (b) and (c) of the definition of metric.

4:F. Prove that for every metric space the Hausdorff distance is a metric in the set

of its closed bounded subsets.


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4. METRIC SPACES 29

LetA and B be bounded polygons in the plane4. Put

d(A, B) =S(A) + S(B) 2S(A B),whereS(C) is the area of polygon C.

4:G. Prove that d is a metric in the set of all plane bounded polygons.

We will call d thearea metric.

4:H. Prove that in the set of all bounded plane polygons the area metric is notequivalent to the Hausdorff metric.

4:I. Prove that in the set of convex bounded plane polygons the area metric is equiv-alent to the Hausdorff metric.

4

17 Asymmetrics

A function : X X R+ is called an asymmetric in setX, if(a) (x, y) = 0 and(y, x) = 0, iffx = y;(b) (x, y) (x, z) + (z, y) for any x, y,z X.

Thus, an asymmetric satisfies the conditions a and c of the definition of metric,but does not satisfy condition b.

An example of asymmetric taken from the real life: the shortest length of pathfrom one point to another by a car in a city in which there exist one way streets.

4:J. Prove that if : X X R+ is an asymmetric then the function(x, y)

(x, y) + (y, x)

is a metric in X.

LetAandBbe bounded subsets of a metric space (X, ). The numbera(A, B) =supbB(b, A) is called the asymmetric distance from AtoB .

4:K. ain the set of nounded subsets of a metric space satisfies the triangle inequalityfrom the definition of asymmetric.

4:L. In a metric space (X, ), a set B is contained in all the closed sets containingA, iffa(A, B) = 0.

4:M. Prove that a is an asymmetric in the set of all bounded closed subsets of ametric space (X, ) .

LetA and B be polygons on the plane. Put

a(A, B) =S(B) S(A B) = S(B A),whereS(C) is the area of polygon C.

4:1. Prove thata is an asymmetric in the set of all planar polygons.

4Although we assume that the notion of bounded polygon is well-known fromelementary geometry, recall the definition. A bounded plane polygon is a set of thepoints of a simple closed polygonal line and the points surrounded by this line. Bya simple closed polygonal line we mean a cyclic sequence of segments such that eachof them starts at the point where the previous one finishes and these are the only

pairwise intersections of the segments.


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4. METRIC SPACES 30

A pair (X, ), where is an asymmetric in X, is called an asymmetric space. Ofcourse, any metric space is an asymmetric space, too. In an asymmetric space, balls

(open and closed) and spheres are defined like in a metric space, see43.4:N. The set of all open balls of an asymmetric space is a base of some topology.

This topology is said to be generatedby the asymmetric.

4:2. Prove that formula a(x, y) = min(x y, 0) defines an asymmetric in[0, ), and that the topology generated by this asymmetric coincides withthe arrow topology, see22.

Proofs and Comments

4.A Indeed, it makes sense to check that all the conditions of thedefinition of metric is satisfied for eachcombination of points x, y z.

4.B Triangle inequality in this case looks as follows|x y| |x z| + |z y|. Put a= x z,b= z y. This turns the triangle inegualityto a well-known inequality|a + b| |a| + |b|.

4.C As in the solution of Problem4.B, the triangle inequality can berewritten as follows:

ni=1(ai+ bi)

2 ni=1 a2i +ni=1 b2i . By twosquaring followed by an obvious simplification, this inequality is reducedto the well-known Cauchy inequality ( aibi)2 a2i b2i .

4.F Show that if d = diam A and a A then A Dd(a). Viceversa: diam Dd(a) 2d (cf. 4.11).

4.G.1 We have to prove that any pointy Br(a,x)(x) belongs toBr(a). In terms of distances, this means that (y, a) < r, if(y, x) 0, x X}is a topologicalstructure. This follows from Lemma 4.G.1 and Theorems 3.Band 3.C.

4.H For this metric, the balls are open intervals. Each open intervalin R appears as a ball. The standard topology in R is defined by the baseconsisting of all open intervals.

4.I If a set contains together with any of its points a ball withcenter at this point, this set is the union of those balls. Thus, it is openin the metric topology. Ifa U, where U is open, then a Br(x) andBr(a,x)(a) Br(x) U, see Lemma 4.G.1.

4.J An indiscrete space does not have enough open sets. Forx, yXandr =(x, y)> 0, the ballDr(x) is not empty and does not coincide

with the whole space.


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4. METRIC SPACES 31

4:A Clearly, the metric in4.A is an ultrametric. The other metricsare not: for each of them you can find points x, y,zsuch that (x, y) =(x, z) + (z, y).

4:B The definition of ultrametric implies that no one of pairwisedistances between points a,b,c can be greater than each of the othertwo.

4:C By4:B, ify Sr(x) and r > s >0 then Bs(y) Sr(x).4:D Let x z = r1

s1p1 , z y = r2

s2p2 and 1 2. Then: x

y = p1

r1s1

+ r2s2

p21

= p1r1s2+r2s1p21

s1s2, hence p(x, y) p1 =

max{(x, z), (z, y)}.

4.L Condition(b, A) = 0 means that each ball centered at b meetsA. In turn, this means that b does not belong to the complement ofA(since Ais closed).


4.2 Cf. 4.B.

4.4 Look for an answer in4.7.

4.7 Squares with sides parallel to the coordinate axes and bisectors of

the coordinate angles, respectively.

4.8 D1(a) = X,D1/2(a) = {a}, S1/2(a) = .4.9 For example,X=D1(0) R1, andD3/2(5/6) D1(0).4.10 Three points suffice.

4.11 Let R > r and DR(b) Dr(a). Take c DR(b) and use thetriangle inequality (b, c) (b, a) + (a, c).

4.12 Put u = b x and t = x a. The Cauchy inequality becomesequality, iff the vectors u and t has the same direction, i.e., x lies on thesegment connecting a andb.

4.13 For metric (p) with p > 1 this set coincides with the segmentconnecting a and b, and for metric (1) it is a rectangular parallelipipedwhose opposite vertices are those two point

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