An Introduction to Metric Spaces

Dhananjay Gopal Associate Professor of Mathematics in Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur (C.G.) India. He was a visiting Professor at the Department of Mathematics, University of Jaen, Spain for the spring semester of 2023. He was Assistant Professor of Applied Mathematics at S.V. National Institute of Technology, Surat, Gujarat, from 2009 to 2020. He has earned his doctorate in Mathematics from Guru Ghasidas University, Bilaspur, India and is currently. His research interest is in the theory of Nonlinear Analysis and Fuzzy Metric Fixed Point Theory. He has authored more than 110 papers in journals, proceedings and three books in the field of metric spaces and fixed point theory. He is an editorial board member of three international journals and a regular reviewer of several journals published by international publishers. He was the guest editor of the special issue " Fixed point theory in abstract metric spaces with generalised contractive conditions; new methods, algorithms, and applications", in the Journal of Mathematics and a Special Issue on “Nonlinear operator theory and its applications” in the Journal of function spaces. D. Gopal has active research collaborations with KMUTT, Bangkok, Thammasat University Bangkok, and Jaen University Spain. Mr. Aniruddha Deshmukh is currently a research scholar in the area of Harmonic Analysis and k-plane transform Group from Indian Institute of Technology, Indore, India 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 a research scholar in the area of Geometry and Topology from Harish-Chandra Research Institute (HRI) Prayagraj (Allahabad), Uttar Pradesh, India 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