Introductory Topology: Exercises And Solutions (Second Edition)

by Mohammed Hichem Mortad

Overview

'The book is well written, and there is a welcome breadth in the choice of topics. I think this book is a valuable resource. Students who meticulously work through all the problems in the book in an intelligent way, will surely gain considerable insight into the subject; teachers who don't tell their students about it will find it a valuable source for exam questions.'
The Mathematical GazetteThe book offers a good introduction to topology through solved exercises. It is mainly intended for undergraduate students. Most exercises are given with detailed solutions.In the second edition, some significant changes have been made, other than the additional exercises. There are also additional proofs (as exercises) of many results in the old section 'What You Need To Know', which has been improved and renamed in the new edition as 'Essential Background'. Indeed, it has been considerably beefed up as it now includes more remarks and results for readers' convenience. The interesting sections 'True or False' and 'Tests' have remained as they were, apart from a very few changes.

Product Details

ISBN-13:	9789813148024
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	12/14/2016
Pages:	376
Product dimensions:	6.00(w) x 9.00(h) x 0.70(d)

Table of Contents

Preface ix

Preface to the Second Edition xiii

Notation and Terminology xv

0.1 Notation xv

0.2 Terminology xvi

Part 1 Exercises 1

Chapter 1 General Notions: Sets, Functions et al. 3

1.1 Essential Background 3

1.1.1 Sets 3

1.1.2 Functions 9

1.1.3 Equivalence Relation 15

1.1.4 Consequences of The Least Upper Bound Property 15

1.1.5 Countability 17

1.2 Exercises With Solutions 19

1.3 More Exercises 24

Chapter 2 Metric Spaces 27

2.1 Essential Background 27

2.1.1 Definitions and Examples 27

2.1.2 Important Sets in Metric Spaces 30

2.1.3 Continuity in Metric Spaces 32

2.1.4 Equivalent Metrics 34

2.2 True or False: Questions 36

2.3 Exercises With Solutions 36

2.4 Tests 42

2.5 More Exercises 43

Chapter 3 Topological Spaces 47

3.1 Essential Background 47

3.1.1 General Notions 47

3.1.2 Separation Axioms 49

3.1.3 Closures, Interiors, Limits Points, et al 50

3.1.4 Bases and Subbases 56

3.1.5 The Subspace Topology 58

3.1.6 The Product and Quotient Topologies 59

3.2 True or False: Questions 61

3.3 Exercises With Solutions 63

3.4 Tests 72

3.5 More Exercises 73

Chapter 4 Continuity and Convergence 77

4.1 Essential Background 77

4.1.1 Continuity 77

4.1.2 Convergence 82

4.1.3 Sequential Continuity 83

4.1.4 A Word on Infinite Product Topology 84

4.2 True or False: Questions 86

4.3 Exorcises With Solutions 87

4.4 Tests 94

4.5 More Exercises 95

Chapter 5 Compact Spaces 99

5.1 Essential Background 99

5.1.1 Compactness: General Notions 99

5.1.2 Compactness and Continuity 102

5.1.3 Sequential Compactness and Total Boundedness 104

5.2 True or False: Questions 106

5.3 Exercises With Solutions 107

5.4 Tests 112

5.5 More Exercises 113

Chapter 6 Connected Spaces 117

6.1 Essential Background 117

6.1.1 Connectedness 117

6.1.2 Components 119

6.1.3 Path-connectedness 120

6.2 True or False: Questions 122

6.3 Exercises With Solutions 123

6.4 Tests 125

6.5 More Exercises 126

Chapter 7 Complete Metric Spaces 129

7.1 Essential Background 129

7.1.1 Completeness 129

7.1.2 Fixed Point Theorem 135

7.1.3 A Word on Integral Equations 137

7.2 True or False: Questions 140

7.3 Exercises With Solutions 141

7.4 Tests 148

7.5 More Exercises 148

Chapter 8 Function Spaces 153

8.1 Essential Background 153

8.1.1 Types of Convergence 153

8.1.2 Weierstraas Approximation Theorem 155

8.1.3 The Arzelà-Ascoli Theorem 155

8.2 True or False: Questions 157

8.3 Exercises With Solutions 157

8.4 Tests 159

8.5 More Exercises 159

Part 2 Solutions 161

Chapter 1 General Notions: Sets, Functions et al 163

1.2 Solutions to Exercises 163

Chapter 2 Metric Spaces 181

2.2 True or False: Answers 181

2.3 Solutions to Exercises 183

2.4 Hints/Answers to Tests 201

Chapter 3 Topological Spaces 203

3.2 True or False: Answers 203

3.3 Solutions to Exercises 209

3.1 Hints/Answers to Tests 241

Chapter 4 Continuity and Convergence 243

4.2 True or False: Answers 243

4.3 Solutions to Exercises 247

4.4 Hints/Answers to Tests 266

Chapter 5 Compact Spaces 269

5.2 True or False: Answers 269

5.3 Solutions to Exercises 272

5.4 Hints/Answers to Tests 293

Chapter 6 Connected Spaces 295

6.2 True or False: Answers 295

6.3 Solutions to Exercises 298

6.4 Hints/Answers to Tests 307

Chapter 7 Complete Metric Spaces 309

7.2 True or False; Answers 309

7.3 Solutions to Exercises 313

7.4 Hints/Answers to Tests 342

Chapter 8 Function Spaces 343

8.2 True or False: Answers 343

8.3 Solutions to Exercises 345

8.4 Hints/Answers to Tests 350

Bibliography 351

Index 353