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1INTRODUCTION TO POINT SET TOPOLOGY 1

1.1 Open and Closed Sets 1

1.2 Continuous Functions 12

1.3 Some Topological Properties 22

1.4 A Brief Introduction to Dimension（Optional） 31

2SURFACES 36

2.1 Definition of a Surface 36

2.2 Connected Sum Construction 39

2.3 Plane Models of Surfaces 41

2.4 Orientability 54

2.5 Plane Models of Nonorientable Surfaces 56

2.6 Classification of Surfaces 57

2.7 Proof of the Classification Theorem for Surfaces（Optional） 59

3THE EULER CHARACTERISTIC 66

3.1 Cell Complexes and the Euler Characteristic 67

3.2 Triangulations 75

3.3 Genus 77

3.4 Regular Complexes 80

3.5 b-Valent Complexes 86

4MAPS AND GRAPHS 90

4.1 Maps and Map Coloring 91

4.2 The Five-Color Theorem for S2 98

4.3 Introduction to Graphs 101

4.4 Graphs in Surfaces 107

4.5 Embedding the Complete Graphs and Graph Coloring 113

5VECTOR FIELDS ON SURFACES 118

5.1 Vector Fields in the Plane 119

5.2 Index of a Critical Point 122

5.3 Limit Sets in the Plane 131

5.4 A Local Description of a Critical Point 133

5.5 Vector Fields on Surfaces 141

6THE FUNDAMENTAL GROUP 152

6.1 Path Homotopy and the Fundamental Group 153

6.2 The Fundamental Group of the Circle 160

6.3 Deformation Retracts 164

6.4 Further Calculations 167

6.5 Presentations of Groups 171

6.6 Seifert-van Kampen Theorem and the Fundamental Groups of Surfaces 173

6.7 Proof of the Seifert-van Kampen Theorem 179

7INTRODUCTION TO KNOTS 182

7.1 Knots：What They Are and How to Draw Them 183

7.2 Prime Knots 190

7.3 Alternating Knots 191

7.4 Reidemeister Moves 193

7.5 Some Simple Knot Invariants 194

7.6 Surfaces with Boundary 202

7.7 Knots and Surfaces 207

7.8 Knot Polynomials 219

Bibliography and Reading List 230

Index 234