elementary topology, a first course - viro, ivanov, netsvetaev, kharlamov

7/29/2019 Elementary Topology, A First Course - Viro, Ivanov, Netsvetaev, Kharlamov

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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 Mathematical

Department 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 know someof these things. Maybe, students studying topology can postpone this material to theirgraduate 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 are ratherisolated from the contents of traditional graduate courses. They are important, butmore related to the material of the very first topology course. In the book these topicsare intertwined with the core material and exercises, but are distinguishable: theyare typeset, like these lines, with large face, large margins, theorems and problems inthem are numerated in a special manner described below.

Exercises and illustrative problems to the additional topics are typesetwith even 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 and slightlysmaller margins as the problems elaborating on the core material.

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

Therefore the book looks like a Russian folklore doll, matreshka com-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. Subsections are not numerated. Each of themis devoted to a single topic and consists of definitions, commentaries,

theorems, exercises, problems, and riddles.


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FOREWORD v

By a riddle we mean a problem of a special sort: its solution is notcontained in the formulation. One has to guess a solution, rather than

deduce 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.

Advice 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 iiiAdvice to the Reader vHow This Book Was Written vii

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 5

16 Subsets 6

17 To Prove Equality of Sets, Prove Inclusions 618 Inclusion Versus Belonging 619 Defining a Set by a Condition 7110 Intersection and Union 7111 Different Differences 9Proofs and Comments

to Statements of the Main Course in 1 10Hints, Comments, Advises, Solutions, and Answers

to Some Problems of1 112. Topology in a Set 12

2

1 Definition of Topological Space 1222 The Simplest Examples 1223 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 15

211 Topology and Arithmetic Progressions 15

212 Neighborhoods 16ix


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CONTENTS x

Proofs and Commentsto Statements of the Main Course in

2 16

Hints, Comments, Advises, Solutions, and Answersto Some Problems of2 17

3. Bases 1931 Definition of Base 1932 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


to Some Problems of3 214. Metric Spaces 23

41 Definition and First Examples 2342 Further Examples 2343 Balls and Spheres 2444 Subspaces of a Metric Space 2445 Surprising Balls 2546 Segments (What Is Between) 25

47 Bounded Sets and Balls 25

48 Norms and Normed Spaces 2549 Metric Topology 26410 Metrizable Topological Spaces 26411 Equivalent Metrics 27412 Ultrametric 27413 Operations with Metrics 28414 Distance Between Point and Set 28415 Distance Between Sets 28416 Distance Between Metric Spaces 29417 Asymmetrics 29Proofs and Commentsto Statements of the Main Course in 4 30Hints, Comments, Advises, Solutions, and Answers

to Some Problems of4 315. Ordered Sets 33

51 Strict Orders 3352 Non-Strict Orders 3353 Relation between Strict and Non-Strict Orders 3454 Cones 3455 Position of an Element with Respect to a Set 35

56 Total Orders 36


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CONTENTS xi

57 Topologies Defined by a Total Order 36

58 Poset Topology 37

59 How to Draw a Poset 38Proofs and Comments


to Some Problems of5 406. Subspaces 42

61 Topology for a subset of a space 4262 Relativity of Openness 4263 Agreement on Notations of Topological Spaces 43Proofs and Comments


to Some Problems of6 447. Position of a Point with Respect to a Set 45

71 Interior, Exterior and Boundary Points 4572 Interior and Exterior 4573 Closure 4574 Frontier 4675 Closure and Interior with Respect to a Finer Topology 4676 Properties of Interior and Closure 46

77 Characterization of Topology by Closure or Interior

Operations 4878 Dense Sets 4879 Nowhere Dense Sets 48710 Limit Points and Isolated Points 49711 Locally Closed Sets 49Proofs and Comments


to Some Problems of7 50

8. Set-Theoretic Digression. Maps 51

8

1 Maps and the Main Classes of Maps 5182 Image and Preimage 5183 Identity and Inclusion 5284 Composition 5285 Inverse and Invertible 5386 Submappings 53

9. Continuous Maps 5491 Definition and Main Properties of Continuous Maps 5492 Reformulations of Definition 5593 More Examples 55

9

4 Behavior of Dense Sets 55


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CONTENTS xii

95 Local Continuity 56

96 Properties of Continuous Functions 56

97 Special About Metric Case 5798 Functions on Cantor Set and Square-Filling Curves 5799 Sets Defined by Systems of Equations and Inequalities 58910 Set-Theoretic Digression. Covers 59911 Fundamental Covers 59

10. Homeomorphisms 61101 Definition and Main Properties of Homeomorphisms 61102 Homeomorphic Spaces 61103 Role of Homeomorphisms 61104 More Examples of Homeomorphisms 6210

5 Examples of Homeomorphic Spaces 63106 Examples of Nonhomeomorphic Spaces 66107 Homeomorphism Problem and Topological Properties 66108 Information (Without Proof) 66109 Embeddings 671010 Information 68

Chapter 2. Topological Properties 69

11. Connectedness 69111 Definitions of Connectedness and First Examples 6911

2 Connected Sets 69113 Properties of Connected Sets 70114 Connected Components 70115 Totally Disconnected Spaces 71116 Frontier and Connectedness 71117 Behavior Under Continuous Maps 71118 Connectedness on Line 72119 Intermediate Value Theorem and Its Genralizations 721110 Dividing Pancakes 731111 Induction on Connectedness 73

1112 Applications to Homeomorphism Problem 74

12. Path-Connectedness 75121 Paths 75122 Path-Connected Spaces 75123 Path-Connected Sets 76124 Path-Connected Components 76125 Path-Connectedness Versus Connectedness 77126 Polygon-Connectedness 77

13. Separation Axioms 79131 Hausdorff Axiom 79

132 Limits of Sequence 79

133 Coincidence Set and Fixed Point Set 80


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CONTENTS xiii

134 Hereditary Properties 80

135 The First Separation Axiom 80

136 The Third Separation Axiom 81137 The Fourth Separation Axiom 81138 Niemytskis Space 82139 Urysohn Lemma and Tietze Theorem 82

14. Countability Axioms 84141 Set-Theoretic Digression. Countability 84142 Second Countability and Separability 84143 Embedding and Metrization Theorems 85144 Bases at a Point 85145 First Countability 8614

6 Sequential Approach to Topology 86147 Sequential Continuity 8715. Compactness 88

151 Definition of Compactness 88152 Terminology Remarks 88153 Compactness in Terms of Closed Sets 89154 Compact Sets 89155 Compact Sets Versus Closed Sets 89156 Compactness and Separation Axioms 90157 Compactness in Euclidean Space 90

158 Compactness and Maps 91

159 Norms in Rn 921510 Closed Maps 92

16. Local Compactness and Paracompactness 93161 Local Compactness 93162 One-Point Compactification 93163 Proper Maps 94164 Locally Finite Collections of Subsets 94165 Paracompact Spaces 95166 Paracompactness and Separation Axioms 95

167 Partitions of Unity 95

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8 Application: Making Embeddings from Pieces 9517. Sequential Compactness 97

171 Sequential Compactness Versus Compactness 97172 In Metric Space 97173 Completeness and Compactness 98174 Non-Compact Balls in Infinite Dimension 98175 p-Adic Numbers 98176 Induction on Compactness 99177 Spaces of Convex Figures 99

Problems for Tests 101


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CONTENTS xiv

Chapter 3. Topological Constructions 103

18. Multiplication 103181 Set-Theoretic Digression. Product of Sets 103182 Product of Topologies 104183 Topological Properties of Projections and Fibers 104184 Cartesian Products of Maps 105185 Properties of Diagonal and Graph 105186 Topological Properties of Products 106187 Representation of Special Spaces as Products 107

19. Quotient Spaces 108191 Set-Theoretic Digression. Partitions and Equivalence

Relations 108

192 Quotient Topology 109193 Topological Properties of Quotient Spaces 109194 Set-Theoretic Digression. Quotients and Maps 110195 Continuity of Quotient Maps 110196 Closed Partitions 111197 Open Partitions 111

20. Zoo of Quotient Spaces 112201 Tool for Identifying a Quotient Space with a Known

Space 112202 Tools for Describing Partitions 11220

3 Entrance to the Zoo 113204 Transitivity of Factorization 114205 Mobius Strip 115206 Contracting Subsets 115207 Further Examples 116208 Klein Bottle 116209 Projective Plane 1172010 You May Have Been Provoked to Perform an Illegal

Operation 1172011 Set-Theoretic Digression. Sums of Sets 117

2012 Sums of Spaces 117

20

13 Attaching Space 1182014 Basic Surfaces 11921. Projective Spaces 121

211 Real Projective Space of Dimension n 121212 Complex Projective Space of Dimension n 121213 Quaternion Projective Spaces and Cayley Plane 122

22. Topological Groups 123221 Algebraic Digression. Groups 123222 Topological Groups 123223 Self-Homeomorphisms Making a Topological Group

Homogeneous 124


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CONTENTS xv

224 Neighborhoods 125

225 Separaion Axioms 125

226 Countability Axioms 125227 Subgroups 126228 Normal Subgroups 127229 Homomorphisms 1272210 Local Isomorphisms 1282211 Direct Products 128

23. Actions of Topological Groups 130231 Actions of Group in Set 130232 Continuous Actions 130233 Orbit Spaces 13023

4 Homogeneous Spaces 13024. Spaces of Continuous Maps 131241 Sets of Continuous Mappings 131242 Topological Structures on Set of Continuous Mappings131243 Topological Properties of Spaces of Continuous Mappings

132244 Metric Case 132245 Interactions With Other Constructions 133246 Mappings X Y Z and X C(Y, Z) 133

Part 2. Algebraic Topology 135

Chapter 4. Fundamental Group and Covering Spaces 136

25. Homotopy 137251 Continuous Deformations of Maps 137252 Homotopy as Map and Family of Maps 137253 Homotopy as Relation 137254 Straight-Line Homotopy 138255 Two Natural Properties of Homotopies 139256 Stationary Homotopy 139

257 Homotopies and Paths 140

258 Homotopy of Paths 14026. Homotopy Properties of Path Multiplication 141

261 Multiplication of Homotopy Classes of Paths 141262 Associativity 141263 Unit 142264 Inverse 142

27. Fundamental Group 144271 Definition of Fundamental Group 144272 Why Index 1? 144

273 High Homotopy Groups 144

274 Circular loops 145


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CONTENTS xvi

275 The Very First Calculations 146

276 Fundamental Group of Product 146

277 Simply-Connectedness 147278 Fundamental Group of a Topological Group 148

28. The Role of Base Point 149281 Overview of the Role of Base Point 149282 Definition of Translation Maps 149283 Properties ofTs 149284 Role of Path 150285 High Homotopy Groups 150286 In Topological Group 150

29. Covering Spaces 152

291 Definition 152292 Local Homeomorphisms Versus Coverings 152293 Number of Sheets 153294 More Examples 153295 Universal Coverings 154296 Theorems on Path Lifting 154297 High-Dimensional Homotopy Groups of Covering Space156

30. Calculations of Fundamental Groups Using UniversalCoverings 157

301 Fundamental Group of Circle 15730

2 Fundamental Group of Pro jective Space 158303 Fundamental Groups of Bouquet of Circles 158304 Algebraic Digression. Free Groups 158305 Universal Covering for Bouquet of Circles 160

31. Fundamental Group and Continuous Maps 162311 Induced Homomorphisms 162312 Fundamental Theorem of High Algebra 163313 Generalization of Intermediate Value Theorem 164314 Winding Number 164315 Borsuk-Ulam Theorem 164

32. Covering Spaces via Fundamental Groups 166321 Homomorphisms Induced by Covering Projections 166322 Number of Sheets 166323 Hierarchy of Coverings 167324 Automorphisms of Covering 167325 Regular Coverings 167326 Existence of Coverings 167327 Lifting Maps 167

Chapter 5. More Applications and Calculations 168

33. Retractions and Fixed Points 168

331 Retractions and Retracts 168


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CONTENTS xvii

332 Fundamental Group and Retractions 169

333 Fixed-Point Property. 169

34. Homotopy Equivalences 171341 Homotopy Equivalence as Map 171342 Homotopy Equivalence as Relation 171343 Deformation Retraction 171344 Examples 172345 Deformation Retraction Versus Homotopy Equivalence172346 Contractible Spaces 173347 Fundamental Group and Homotopy Equivalences 173

35. Cellular Spaces 175

351 Definition of Cellular Spaces 175

352 First Examples 177353 More Two-Dimensional Examples 178354 Topological Properties of Cellular Spaces 179355 Embedding to Euclidean Space 179356 One-Dimensional Cellular Spaces 180357 Euler Characteristic 181

36. Fundamental Group of a Cellular Space 182361 One-Dimensional Cellular Spaces 182362 Generators 182363 Relators 18236

4 Writing Down Generators and Relators 183365 Fundamental Groups of Basic Surfaces 184366 Seifert - van Kampen Theorem 185

37. One-Dimensional Homology and Cohomology 186371 Description ofH1(X) in Terms of Free Circular Loops 186372 One-Dimensional Cohomology 187373 Cohomology and Classification of Regular Coverings 188374 Integer Cohomology and Maps to S1 188375 One-Dimensional Homology Modulo 2 188

Part 3. Manifolds 190

Chapter 6. Bare Manifolds 192

38. Locally Euclidean Spaces 192381 Definition of Locally Euclidean Space 192382 Dimension 192383 Interior and Boundary 193

39. Manifolds 196391 Definition of Manifold 196392 Components of Manifold 196

393 Making New Manifolds out of Old Ones 196

394 Double 197


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CONTENTS xviii

395 Collars and Bites 197

40. Isotopy 199401 Isotopy of Homeomorphisms 199402 Isotopy of Embeddings and Sets 199403 Isotopies and Attaching 200404 Connected Sums 201

41. One-Dimensional Manifolds 202411 Zero-Dimensional Manifolds 202412 Reduction to Connected Manifolds 202413 Examples 202414 Statements of Main Theorems 202

415 Lemma on 1-Manifold Covered with Two Lines 203

416 Without Boundary 204417 With Boundary 204418 Consequences of Classification 204419 Mapping Class Groups 204

42. Two-Dimensional Manifolds 205421 Examples 205422 Ends and Odds 205423 Closed Surfaces 206424 Triangulations of Surfaces 207425 Two Properties of Triangulations of Surfaces 20742

6 Scheme of Triangulation 208427 Examples 208428 Families of Polygons 209429 Operations on Family of Polygons 2104210 Topological and Homotopy Classification of Closed

Surfaces 2114211 Recognizing Closed Surfaces 2124212 Orientations 2124213 More About Recognizing Closed Surfaces 2134214 Compact Surfaces with Boundary 213

4215 Simply Connected Surfaces 213

43. One-Dimensional mod2-Homology of Surfaces 214431 Polygonal Paths on Surface 214432 Subdivisions of Triangulation 214433 Bringing Loops to General Position 215434 Cutting Surface Along Curve 216435 Curves on Surfaces and Two-Fold Coverings 217436 One-Dimensional Z2-Cohomology of Surface 217437 One-Dimensional Z2-Homology of Surface 218438 Poincare Duality 218439 One-Sided and Two-Sided Simple Closed Curves on

Surfaces 218


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CONTENTS xix

4310 Orientation Covering and First Stiefel-Whitney Class 218

4311 Relative Homology 218

44. Surfaces Beyond Classification 219441 Genus of Surface 219442 Systems of disjoint curves on a surface 219443 Polygonal Jordan and Schonflies Theorems 219444 Polygonal Annulus Theorem 219445 Dehn Twists 219446 Coverings of Surfaces 219447 Branched Coverings 219448 Mapping Class Group of Torus 219449 Braid Groups 219

45. Three-Dimensional Manifolds 220451 Poincare Conjecture 220452 Lens Spaces 220453 Seifert Manifolds 220454 Fibrations over Circle 220455 Heegaard Splitting and Diagrams 220


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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: the

language of set-theoretic topology, which treats the basic notions relatedto continuity. The term general topology means: 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 we

understand 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.

If x is an element of a set A, we write x A and say x belongs to Aand A contains x. The sign

is a version of Greek letter epsilon, which

is the first letter of the Latin word element. To make formulas more3


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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 the

inequality symbols < and > is emphasized. To state that x is not anelement of A, 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 word set has 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 words set and element are 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 an infinitesimal 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 set and 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

by R. 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, x andA, 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 and B are sets and every element of A belongs also to B, we saythat A is a subset of B, 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 set A consists ofa elements, and a set B ofb elements. Provethat if A B then a b.

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

and

.

1.B Reflexivity of Inclusion. Any set includes itself: A A holdstrue for any A.

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

1.C The Empty Set Is Everywhere. A for 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 Aitself. A subset of A different from and A is called a proper subset of

A. This word is used when one does not want to consider the obvioussubsets (which are called improper).

1.D Transitivity of Inclusion. If A, B and C are sets, A B andB C, then A C.

17 To Prove Equality of Sets, Prove InclusionsWorking with sets, we need from time to time to prove that two sets,

say A 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 if A B and B A.

18 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, A B means that A is one of the elements of B(that is one of indivisible pieces comprising B), while A

B means that

A is made of some of the elements of B.


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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 suchthat A 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.

19 Defining a Set by a ConditionAs we know (see Section 15), 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 ofsome 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 < 5}, (b) {x N | x < 0}, (c) {x Z | x < 0}.

110 Intersection and UnionThe intersection of sets A and B is the set consisting of their common

elements, that is elements belonging both to A and B. It is denoted byA B and can be described by formula

A B = {x | x A x B}.

Sets A and B are said to be disjoint, if their intersection is empty, i.e.,A B = .

The union of sets A and B is the set consisting of those elements eachof which belongs to at least one of these sets. The union of A and B isdenoted by A B. It can be described by formula

A B = {x | x A or x B}.


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

A or x

B means that x belongs to at least one of the

sets A and 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 A and B

A B = B A A B = B A.1.6. Prove that for any set A

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

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 intersection of the sets belongingto is the set formed by elements which belong to every set, belongingto . This set is denoted by AA 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 by AA 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 sets A, B and C

(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) C of 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 called Venn 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 Theorem 1.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).

111 Different DifferencesA difference A B of sets A and B is the set of those elements of A

which do not belong to B. Here it is not assumed that A B.If A B, the set A B is called also the complement of B in A.

1.10. Prove that for any sets A and B their union A 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 that A (A B) = A B for any sets A and B.1.12. Prove that A B, if and only if A B = .1.13. Prove that A (B C) = (A B) (A C) for any sets A, B and C.

The set (A B) (B A) is called the symmetric difference of 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 A and B.

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

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

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

1.17 Distributivity. Prove that (A B)

C = (A

C) (B

C) for anysets A, 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 Commentsto Statements of the Main Course in 1

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 thatA consists ofa elements? It means, say, that we can count elements of Aone 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 of B with counting the elements of A.The counting of elements of A 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 elements

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

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

1.C Recall that, by the definition of inclusion, A B means thateach element of A is an element of B. Thus we need to prove that anyelement of belongs 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 can

it happen that is not a subset ofA? It could happen, only if there was


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an element of which would not be an element of A. But there is nosuch an element in , because has no elements at all.

1.D We have to prove that each element of A is an element of C.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 that A A. Hence ifA = B then A Band 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 Y of all the sets X such that X X? Does Y belongs toitself? IfY Y then Y Y, since each element X ofY has the propertythat X X. If Y Y then Y Y since Y is the set of ALL the setsX such that X X. This contradiction shows that our definition of Ydoes 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 Take A = {1}, B = {{1}} and C = {{{1}}}. It is more difficultto construct sets A, B and C such that A B, B C, and A C. TakeA = {1}, B = {{1}}, C = {{1}, {{1}}}.

Hints, Comments, Advises, Solutions, and Answersto Some Problems of 1

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 = 1

2and 1 if x = 1

2.

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 X be 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 set and the whole X belong to .

Then is called a topological structure or just a topology1 in X;

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

an element of X is called a point of this topological space; an element of is called an open set of 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 topologicalstructure hold true.

An indiscrete topological space is the opposite example, in which thetopological structure is the most meager. It consists only of X and .

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. Let X be a plane. Let consist of , X, and all open disks with centerat the origin. Is this a topological structure?

2.3. Let X consist 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 of 2.1 is called an arrow. We denote the space of 2.3 (a) by .It is a sort of toy space made of 4 points. Both of these spaces, as well as the

the arrow:

space of 2.2, 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 X be the set R of 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 standard topology 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 of R. Is a topological structure?

The space of 2.5 is denoted by RT1 and called the line with T1-topology.

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

{{a} U : U } {}a topological structure in Y?

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 Problem 2.7 is 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 of X are calledpoints, and elements of are called open sets.2

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

A set F X is said to be closed in 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) X A

A =A

(X A)

(4) X A

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 SetLet K be 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, K is 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 of K.

2:A.1. Prove that(a) K is contained in [0, 1],(b) K does not intersect

13 ,

23

,

(c) K does not intersect3s+13k

, 3s+23k

for any integers k and s.

2:A.2. Present K as [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 collection F of subsets of X satisfies the followingconditions:(a) the intersection of any family of sets from Fbelongs to F;(b) the union of any finite number sets from Fbelongs to F;(c) and X belong to F,

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 subset F of 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 Commentsto Statements of the Main Course in 2

2.A What should we check? The first axiom reads here that theunion of any collection of subsets of X is a subset of X? Well, this isright. IfA X for each A then AA X. Indeed, take arbitrarypoint b

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

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

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 spaceare both open and closed. In a discrete space any set is both open and

closed. 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 : sets X, , {b,c,d}, {a,c,d}, {b, d}, {d}, {c, d};(e) in RT1 : 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=1

0; 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 A B 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 collection of open sets is a base for the topology, iff for any openset U and any point x U there is a set V such that x V U.3.B. A collection of subsets of a set X is a base for some topology in

X, iff X 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 of R2:

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 subbase for , 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 X a 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 X such that 1 2

then 2 is said to be finer than 1, and 1 coarser than 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 Section 2) is coarser than the canonicaltopology in the real line.3.12. Enigma. Let 1 and 2 be bases for topological structures 1 and 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 be equivalent.

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


3

3.A Let be a base of and U . Present U as a union ofelements of . Each point x U is 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 U choose according to theassumption a set Vx

such that x

Vx

U and consider

xUVx.

Notice that xUVx U, since Vx U for each x U. On the otherhand, each point x U is contained in its Vx and hence in xUVx.Therefore U xUVx. Thus, U = xUVx.

3.B Assume that is a base of a topology. Then X, 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 of X such thatX is a union of sets belonging to and the intersection of any two sets

belonging to is a union of sets belonging to . Let us prove that theset 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 B can 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, X is a union of sets belonging to .

3.D Let 1 and 2 be bases of topological structures 1 and 2 in aset X. Obviously, 1 2, iff U 1 x 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


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3. BASES 22

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

space.(d) In the arrow {[0, +), (r, +)}rQ+ is a base.

3.3 The whole topological structure is its own base. So, the question iswhen 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 the

standard topology of the line gives a base of the same topology! Let U be 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 needat 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 U and 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 a

topological structure. 1 is a base of some topology by 3.B and 3.6. So, youneed to prove analogues of 3.6 for 2 and . To prove that the structuresdefined, say, by bases 1 and 2, you need to prove that a union of disks canbe 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 concerning 3.5 and 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, iff x = 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, if x = y;1, if x = 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.

Metrics 4.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 in 4.C4.2 are included in infinite series of themetrics

(p) : (x, y) ni=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/qif xi, yi 0, p, q > 0 and 1p + 1q = 1.

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

limp+

ni=1

api

1p

= max ai,

for any positive a1, a2, . . . , an.

23


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4. METRIC SPACES 24

4.4. Enigma. How is this related to 2, , and 1 from Section 3?

For a real number p 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 series i=1 |xi yi|pconverges and that

(x, y) i=1

|xi yi|p1p

, p 1

is a metric in l(p).

43 Balls and Spheres

Let (X, ) be a metric space, let a be its point, and let r be a positivereal 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 SpaceIf (X, ) is a metric space and A X, then the restriction of metric

to A A is 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 (n 1)-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 sphere and ball (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 if r < (x, a)? What is an analogue for the statement

of Problem 4.E in 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 of 4.1 and 4.2 (cf. 4.4)?

4.8. Find D1(a), D 12

(a), and S12

(a) in the space of 4.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.)

47 Bounded Sets and Balls

A subset A of a metric space (X, ) is said to be bounded, if there is anumber d > 0 such that (x, y) < d for any x, y A. The greatest lowerbound of such d is called the diameter of A and 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 Spaces

Let X be a vector space (over R). Function X R+ : x ||x|| is calleda norm if(a) ||x|| = 0, iff x = 0;(b) ||x|| = ||||x|| for any R and x X;(c) ||x + y|| ||x|| + ||y|| for any x, y X.

4.15. Prove that if x ||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 called metric topology. It is said to be induced by 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 inSection 2 is induced by metric (x, y) |x y|.

4.19. What topological structure is induced by the metric of 4.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.

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.

410 Metrizable Topological Spaces

A topological space is said to be metrizable if 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 Section 2 are metrizable?3Recall that a set A is said to be convex if for any x, y A the segment connecting

x, 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 Equivalent Metrics

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

4.26. Are the metrics of 4.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 4.27 can be weakened. How?

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

4.31*. Prove that the following two metrics 1, C in the set of all contin-uous functions [0, 1] R are not equivalent:4

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?

412 UltrametricA metric is called an ultrametric if 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.A4.2 is ultrametric. Which one?

4:B. Prove that in an ultrametric space all triangles are isosceles (i.e., for any threepoints a, 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 alsoopen.

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 differencex y as rsp, where r, s, and are integers, and r, s are relatively prime with p. Put(x, y) = p.

4:D. Prove that this is an ultrametric.

4Indexes in the notations allude to the spaces these metrics are defining.


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4. METRIC SPACES 28

413 Operations with Metrics

4.32. Prove that if 1, 2 are metrics in X then 1 + 2 and max{1, 2} arealso metrics. Are the functions min{1, 2}, 1

2, and 12 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, if f satisfies the following conditions:

(1) f(0) = 0,

(2) f is 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.

414 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 point b to the set A and denoted by (b, A).

4.L. Let A be 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.

415 Distance Between SetsLet A 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 between A and B.

4.36. Prove that the Hausdorff distance in the set of all bounded subsets ofa metric space satisfies the conditions (b) and (c) of the definition of metric.

4.37. Prove that for every metric space the Hausdorff distance is a metric inthe set of its closed bounded subsets.

Let A and B be bounded polygons in the plane5. Put

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

5Although 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 29

4.38. Prove that d is a metric in the set of all plane bounded polygons.

We will call d the area metric.4.39. Prove that in the set of all bounded plane polygons the area metric isnot equivalent to the Hausdorff metric.

4.40. Prove that in the set of convex bounded plane polygons the area metricis equivalent to the Hausdorff metric.

416 Distance Between Metric SpacesWrite about Gromov distance!

417 Asymmetrics

A function : X X R+ is called an asymmetric in set X, if(a) (x, y) = 0 and (y, x) = 0, iff x = 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:E. Prove that if : X X R+ is an asymmetric then the function(x, y) (x, y) + (y, x)

is a metric in X.

Let A and B be bounded subsets of a metric space (X, ). The number a(A, B) =supbB (b, A) is called the asymmetric distance from A to B.

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

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

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

Let A and B be polygons on the plane. Put

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

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

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, see 43.4:I. The set of all open balls of an asymmetric space is a base of some topology.

This topology is said to be generated by 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 with

the arrow topology, see 2

2.


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4. METRIC SPACES 30


4

4.A Indeed, it makes sense to check that all the conditions of thedefinition of metric is satisfied for each combination 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 Problem 4.B, the triangle inequality can berewritten as follows:

ni=1(ai + bi)

2

ni=1 a

2i +

ni=1 b

2i . By two

squaring followed by an obvious simplification, this inequality is reduced

to the well-known Cauchy inequality ( aibi)2 a2i b2i .4.F Show that if d = diam A and a A then A Dd(a). Vice

versa: diam Dd(a) 2d (cf. 4.11).4.G.1 We have to prove that any point y Br(a,x)(x) belongs to

Br(a). In terms of distances, this means that (y, a) < r, if (y, x) 0, x

X

}is a topological

structure. This follows from Lemma 4.G.1 and Theorems 3.B and 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. If a 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. For x, y X and r = (x, y) > 0, the ball Dr(x) is not empty and does not coincidewith the whole space.

4:A Clearly, the metric in 4.A is an ultrametric. The other metricsare not: for each of them you can find points x,y,z such 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 By 4:B, if y Sr(x) and r > s > 0 then Bs(y) Sr(x).


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4. METRIC SPACES 31

4:D Let x z = r1s1

p1 , z y = r2s2

p2 and 1 2. Then: x

y = p

1 r1s1 + r2s2p21 = p1 r1s2+r2s1p21s1s2 , hence p(x, y) p1 =max{(x, z), (z, y)}.4.L Condition (b, A) = 0 means that each ball centered at b meets

A. In turn, this means that b does not belong to the complement of A(since A is closed).


4.2 Cf. 4.B.

4.4 Look for an answer in 4.7.

4.7 Squares with sides parallel to the coordinate axes and bisectors ofthe coordinate angles, respectively.

4.8 D1(a) = X, D1/2(a) = {a}, S1/2(a) = .4.9 For example, X = D1(0) R1

elementary topology, a first course - viro, ivanov, netsvetaev, kharlamov

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