Principles of topology / Fred H. Croom, The University of the South.

Format:

Book

Author/Creator:

Croom, Fred H., 1941-

Language:

English

Subjects (All):

Topology.

Physical Description:

xi, 312 pages ; 24 cm

Edition:

Dover edition.

Place of Publication:

Mineola, New York : Dover Publications, Inc., 2016.

Summary:

Topology is a natural, geometric, and intuitively appealing branch of mathematics that can be understood and appreciated by students as they begin their study of advanced mathematical topics. Designed for a one-semester introduction to topology at the undergraduate and beginning graduate levels, this text is accessible to students familiar with multivariable calculus. Rigorous but not abstract, the treatment emphasizes the geometric nature of the subject and the applications of topological ideas to geometry and mathematical analysis. Customary topics of point-settopology include metric spaces, general topological spaces, continuity, topological equivalence, basis, subbasis, connectedness, compactness, separation properties, metrization, subspaces, product spaces, and quotient spaces. In addition, the text introduces geometric, differential, and algebraic topology. Each chapter includes historical notes to put important developments into their historical framework. Exercises of varying degrees of difficulty form an essential part of the text. Dover (2016) republication of the edition originally published by Saunders College Publishing, Philadelphia, 1989, and by Cengage Learning Asia, 2002. Book jacket.

Contents:

Chapter 1 Introduction 1

1.1 The Nature of Topology 1

1.2 The Origin of Topology 7

1.3 Preliminary Ideas from Set Theory 11

1.4 Operations on Sets: Union, Intersection, and Difference 14

1.5 Cartesian Products 19

1.6 Functions 20

1.7 Equivalence Relations 25

Chapter 2 The Line and the Plane 29

2.1 Upper and Lower Bounds 29

2.2 Finite and Infinite Sets 33

2.3 Open Sets and Closed Sets on the Real Line 39

2.4 The Nested Intervals Theorem 46

2.5 The Plane 49

Suggestions for Further Reading 51

Historical Notes for Chapter 2 53

Chapter 3 Metric Spaces 55

3.1 The Definition and Some Examples 55

3.2 Open Sets and Closed Sets in Metric Spaces 61

3.3 Interior, Closure, and Boundary 69

3.4 Continuous Functions 75

3.5 Equivalence of Metric Spaces 78

3.6 New Spaces from Old 82

3.7 Complete Metric Spaces 87

Suggestions for Further Reading 96

Historical Notes for Chapter 3 97

Chapter 4 Topological Spaces 99

4.1 The Definition and Some Examples 99

4.2 Interior, Closure, and Boundary 103

4.3 Basis and Subbasis 109

4.4 Continuity and Topological Equivalence 115

4.5 Subspaces 122

Suggestions for Further Reading 128

Historical Notes for Chapter 4 129

Chapter 5 Connectedness 131

5.1 Connected and Disconnected Spaces 131

5.2 Theorems on Connectedness 133

5.3 Connected Subsets of the Real Line 143

5.4 Applications of Connectedness 144

5.5 Path Connected Spaces 147

5.6 Locally Connected and Locally Path Connected Spaces 154

Suggestions for Further Reading 159

Historical Notes for Chapter 5 160

Chapter 6 Compactness 161

6.1 Compact Spaces and Subspaces 161

6.2 Compactness and Continuity 167

6.3 Properties Related to Compactness 172

6.4 One-Point Compactification 181

6.5 The Cantor Set 187

Suggestions for Further Reading 192

Historical Notes for Chapter 6 193

Chapter 7 Product and Quotient Spaces 195

7.1 Finite Products 195

7.2 Arbitrary Products 204

7.3 Comparison of Topologies 211

7.4 Quotient Spaces 213

7.5 Surfaces and Manifolds 221

Suggestions for Further Reading 228

Historical Notes for Chapter 7 229

Chapter 8 Separation Properties and Metrization 231

8.1 T₀, T₁, and T₂-Spaces 231

8.2 Regular Spaces 234

8.3 Normal Spaces 237

8.4 Separation by Continuous Functions 243

8.5 Metrization 253

8.6 The Stone-Cech Compactification 261

Suggestions for Further Reading 265

Historical Notes for Chapter 8 266

Chapter 9 The Fundamental Group 267

9.1 The Nature of Algebraic Topology 267

9.2 The Fundamental Group 267

9.3 The Fundamental Group of S¹ 280

9.4 Additional Examples of Fundamental Groups 286

9.5 The Brouwer Fixed Point Theorem and Related Results 291

9.6 Categories and Functors 295

Suggestions for Further Reading 298

Historical Notes for Chapter 9 299.

Notes:

Originally published: Philadelphia : Saunders College Publishing, 1989; slightly corrected.

Includes bibliographical references and index.

ISBN:

9780486801544

0486801543

OCLC:

916408939

Publisher Number:

99968487036