An Introduction to Metric Spaces

Dr. Dhananjay Gopal has a doctorate in Mathematics from Guru Ghasidas University, Bilaspur, India, and is currently Assistant Professor of Applied Mathematics in S V National Institute of Technology, Surat, Gujarat, India. He is author and co-author of several papers in journals, proceedings, and a monograph on Background and Recent Developments of Metric Fixed Point Theory. He is devoted to general research on the theory of Nonlinear Analysis and Fuzzy Metric Fixed Point Theory. Mr. Aniruddha Deshmukh is currently a student of (Integrated) MSc Mathematics and is associated to the Applied Mathematics and Humanities Department, S V National Institute of Technology, Surat, Gujarat, India. He has been an active student in the department and has initiated many activities for the benefit of the students, especially as a member of the science community (student chapter), known by the name of SCOSH. During his course, he has also attended various internships and workshop such as the Mathematics Training and Talent Search (MTTS) Programme for two consecutive years (2017–2018) and has also done a project on the qualitative questions on Differential Equations at Indian Institute of Technology (IIT), Gandhinagar, Gujarat, India in 2019. He has also qualified CSIR-NET JRF. Furthermore, his research interest focuses on Linear Algebra and Analysis and their applicability in solving some real-world problems. Abhay S. Ranadive is a Professor at the Department of Pure & Applied Mathematics Ghasidas Vishwavidyalaya (A Central University), Bilaspur, Chattisgarh, India. He has been teaching at the university for the last 30 years. He is author and co-author of several papers in journals and proceedings. He is devoted to general research on the theory of fuzzy sets and fuzzy logic, modules, and metric fixed point. Mr. Shubham Yadav is currently a student of (Integrated) M.Sc. Mathematics and is associated to the Applied Mathematics and Humanities Department, S V National Institute of Technology, Surat, Gujarat, India. As a member of SCOSH the student prominent science community in the institute, he has attended and organized various workshops and seminars. He also attended Madhava Mathematics Camp 2017. He did an internship on the calculus of fuzzy numbers at NIT, Trichy and one on operator theory at IIT, Hyderabad. He has also qualified for JRF. His main research interests are functional analysis and fuzzy sets with a knack for learning abstract mathematical concepts.

Contents

Preface ix

A Note to the Reader xiii

Authors xv

1 Set Theory 1

1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The empty set . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Operations on sets . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Uniqueness of the empty set . . . . . . . . . . . . . . . 9

1.1.4 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.5 Cartesian products . . . . . . . . . . . . . . . . . . . . 9

1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Types of relations . . . . . . . . . . . . . . . . . . . . 12

1.2.2 Equivalence relations . . . . . . . . . . . . . . . . . . . 13

1.2.3 Partition of sets . . . . . . . . . . . . . . . . . . . . . 15

1.2.4 Partial order relations . . . . . . . . . . . . . . . . . . 16

1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.1 Composition of functions . . . . . . . . . . . . . . . . 24

1.3.2 Inverse of a function . . . . . . . . . . . . . . . . . . . 26

1.3.3 Images of sets under functions . . . . . . . . . . . . . 32

1.3.4 Inverse images of sets under functions . . . . . . . . . 36

1.4 Countability of Sets . . . . . . . . . . . . . . . . . . . . . . . 39

1.4.1 Finite sets . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.4.2 Countable sets . . . . . . . . . . . . . . . . . . . . . . 44

Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2 Metric Spaces 55

2.1 Review of Real Number System and Absolute Value . . . . . 55

2.2 Young, H¨older, andMinkowski Inequalities . . . . . . . . . . 57

2.3 Notion ofMetric Space . . . . . . . . . . . . . . . . . . . . . 64

2.4 Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.4.1 Subspace topology . . . . . . . . . . . . . . . . . . . . 96

2.4.2 Product topology . . . . . . . . . . . . . . . . . . . . . 97

2.5 Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

2.6 Interior, Exterior, and Boundary Points . . . . . . . . . . . . 101

2.7 Limit and Cluster Points . . . . . . . . . . . . . . . . . . . . 104

2.8 Bounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.9 Distance Between Sets . . . . . . . . . . . . . . . . . . . . . 112

2.10 EquivalentMetrics . . . . . . . . . . . . . . . . . . . . . . . . 115

Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3 Complete Metric Spaces 129

3.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.1.1 Subsequences . . . . . . . . . . . . . . . . . . . . . . . 130

3.2 Convergence of Sequence . . . . . . . . . . . . . . . . . . . . 131

3.3 CompleteMetric Spaces . . . . . . . . . . . . . . . . . . . . . 139

3.4 Completion ofMetric Spaces . . . . . . . . . . . . . . . . . . 143

3.4.1 Construction of the set Z . . . . . . . . . . . . . . . . 145

3.4.2 Embedding X in Z . . . . . . . . . . . . . . . . . . . . 147

3.4.3 Proving Z is complete . . . . . . . . . . . . . . . . . . 147

3.4.4 Uniqueness of extension up to isometry . . . . . . . . 148

3.5 Baire Category Theorem . . . . . . . . . . . . . . . . . . . . 149

3.5.1 Category of sets . . . . . . . . . . . . . . . . . . . . . 149

3.5.2 Baire category theorem . . . . . . . . . . . . . . . . . 151

3.5.3 Applications of Baire category theorem . . . . . . . . 153

Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 158

4 Compact Metric Spaces 161

4.1 Open Cover and Compact Sets . . . . . . . . . . . . . . . . . 161

4.2 General Properties of Compact Sets . . . . . . . . . . . . . . 165

4.3 Sufficient Conditions for Compactness . . . . . . . . . . . . . 169

4.4 Sequential Compactness . . . . . . . . . . . . . . . . . . . . . 172

4.5 Compactness: Characterizations . . . . . . . . . . . . . . . . 174

Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5 Connected Spaces 183

5.1 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . 183

5.1.1 Connected subsets . . . . . . . . . . . . . . . . . . . . 185

5.2 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.3 Totally Disconnected Spaces . . . . . . . . . . . . . . . . . . 192

Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6 Continuity 195

6.1 Continuity of Real Valued Functions . . . . . . . . . . . . . . 195

6.2 Continuous Functions in ArbitraryMetric Spaces . . . . . . 197

6.2.1 Equivalent definitions of continuity and other

characterizations . . . . . . . . . . . . . . . . . . . . . 202

6.2.2 Results on continuity . . . . . . . . . . . . . . . . . . . 210

6.3 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . 217

6.4 Continuous Functions on Compact Spaces . . . . . . . . . . . 224

6.5 Continuous Functions on Connected Spaces . . . . . . . . . . 229

6.5.1 Path connectedness . . . . . . . . . . . . . . . . . . . . 237

6.6 Equicontinuity and Arzela-Ascoli’s Theorem . . . . . . . . . 242

6.7 Open and ClosedMaps . . . . . . . . . . . . . . . . . . . . . 245

6.8 Homeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . 246

Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 252

7 Banach Fixed Point Theorem and Its Applications 255

7.1 Banach Contraction Theorem . . . . . . . . . . . . . . . . . 255

7.2 Applications of Banach Contraction Principle . . . . . . . . . 260

7.2.1 Root finding problem . . . . . . . . . . . . . . . . . . 260

7.2.2 Solution of systemof linear algebraic equations . . . . 261

7.2.3 Picard existence theorem for differential equations . . 264

7.2.4 Solutions of integral equations . . . . . . . . . . . . . 267

7.2.5 Solutions of initial value and boundary value

problems . . . . . . . . . . . . . . . . . . . . . . . . . 271

7.2.6 Implicit function theorem . . . . . . . . . . . . . . . . 273

Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 276

Appendix A 277

Bibliography 281

Index 283