Table of contents for Beginning topology / Sue E. Goodman.


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1
INTRODUCTION 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
2
SURFACES 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
3
THE 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
4
MAPS 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
5
VECTOR 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
6
THE 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
7
INTRODUCTION 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



Library of Congress subject headings for this publication: Topology Textbooks