A Comprehensive Textbook on Metric Spaces

This textbook provides a comprehensive course in metric spaces. Presenting a smooth takeoff from basic real analysis to metric spaces, every chapter of the book presents a single concept, which is further unfolded and elaborated through related sections and subsections. Apart from a unique new presentation and being a comprehensive textbook on metric spaces, it contains some special concepts and new proofs of old results, which are not available in any other book on metric spaces. It has individual chapters on homeomorphisms and the Cantor set. This book is almost self-contained and has an abundance of examples, exercises, references and remarks about the history of basic notions and results. Every chapter of this book includes brief hints and solutions to selected exercises. It is targeted to serve as a textbook for advanced undergraduate and beginning graduate students of mathematics. Foreword Preface Contents 1 Real Analysis 1.1 The Real Number System 1.2 Sequences of Real Numbers 1.2.1 Convergence of a Sequence 1.2.2 Algebra of Limits 1.2.3 Bounded Monotone Sequences 1.2.4 Cauchy Sequences 1.3 Series Convergence 1.4 Decimal and General Expansions 1.5 Continuity 1.6 Uniform Convergence 1.6.1 Necessary and Sufficient Conditions 1.6.2 Notes and Remarks 1.7 Hints and Solutions to Selected Exercises References 2 Metric Spaces 2.1 Introduction 2.1.1 The Euclidean Spaces 2.1.2 Balls and Bounded Sets 2.2 Convergence in Metric Spaces 2.3 Normed Linear Spaces 2.4 Sequence Spaces 2.5 Hints and Solutions to Selected Exercises References 3 Topology 3.1 Open Sets and Closed Sets 3.2 Limit Points and Isolated Points 3.3 Closures and Boundaries 3.4 Subspace Topology 3.5 Limits and Continuity 3.5.1 The Case of Euclidean Spaces 3.5.2 Continuity and Uniform Convergence 3.6 Topology of Normed Linear Spaces 3.7 Hints and Solutions to Selected Exercises References 4 Completeness 4.1 Introduction 4.2 Banach Contraction Principle 4.3 Characterizations of Completeness 4.3.1 Cantor Intersection Property 4.3.2 Totally Bounded Sets 4.4 Completion of a Metric Space 4.5 Banach Spaces 4.6 Hints and Solutions to Selected Exercises References 5 Compactness 5.1 Introduction 5.1.1 Compact Sets and Closed Sets 5.1.2 Compact Subsets of Euclidean Spaces 5.2 Characterizations of Compact Sets 5.2.1 Finite Intersection Property 5.2.2 Sequentially Compact Sets 5.3 Continuity and Compactness 5.3.1 Uniform Continuity 5.3.2 Notes and Remarks 5.4 Lipschitz Continuity 5.5 Hints and Solutions to Selected Exercises References 6 Connectedness 6.1 Path Connectedness 6.2 Connected Sets 6.3 Components 6.4 Miscellaneous 6.4.1 Locally Connected and Locally Path Connected Spaces 6.4.2 Path Connectedness in Locally Path Connected Spaces 6.4.3 Quasi-components 6.4.4 Totally Disconnected Sets 6.5 Hints and Solutions to Selected Exercises References 7 Cardinality 7.1 Countable and Uncountable Sets 7.2 Some Applications to Topology 7.3 The Set of Discontinuities 7.3.1 The Case of Monotone Functions 7.3.2 The General Case 7.4 Cardinality 7.4.1 Cardinal Numbers 7.4.2 Notes and Remarks 7.5 Hints and Solutions to Selected Exercises References 8 Denseness 8.1 Separability 8.2 Perfect Sets 8.3 Baire Category Theorem 8.4 Equicontinuity 8.5 Hints and Solutions to Selected Exercises References 9 Homeomorphisms 9.1 Equivalent Metrics 9.2 Homeomorphisms 9.3 Extension Theorems for Continuous Functions 9.4 Finite-Dimensional Normed Linear Spaces 9.5 Hints and Solutions to Selected Exercises References 10 The Cantor Set 10.1 Introduction 10.2 An Infinite Product Representation 10.3 Embedding Cantor Set Inside Metric Spaces 10.4 Characterizations in Terms of the Cantor Set 10.4.1 Cantor Set and Compact Metric Spaces 10.4.2 Cantor Set and Totally Disconnected Metric Spaces 10.4.3 Open Subsets of the Cantor Set 10.5 Miscellaneous 10.5.1 The Cantor Function 10.5.2 Homeomorphic Permutations of the Cantor Set 10.5.3 Cantor's Leaky Tent 10.6 Hints and Solutions to Selected Exercises References Appendix A Axiomatic Set Theory A.1 The Language of Set Theory A.2 Zermelo-Fraenkel Axioms A.3 The Set of Non-Negative Integers A.4 Axiom of Choice A.5 Hints and Solutions to Selected Exercises Appendix B More on Continuous Functions B.1 Weierstrass Approximation Theorem B.2 A Continuous but Nowhere Differentiable Function B.3 Most Continuous Functions are Nowhere Differentiable Appendix C Proofs Through Games C.1 An Infinite Game and Uncountable Sets C.2 The Banach-Mazur Game Appendix D A Glimpse into General Topology D.1 Introduction to Topological Spaces D.2 Analogies and Contrasts References Index