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With a nice balance of mathematical precision and accessibility, this text provides a broad introduction to the field of topology. Author Sue Goodman piques student curiosity and interest without losing necessary rigor so that they can appreciate the beauty and fun of mathematics. The text demonstrates that mathematics is an active and ever-changing field with many problems still unsolved, and students will see how the various areas of mathematics-algebra, combinatorics, geometry, calculus, and differential equations-interact with topology. Students learn some of the major ideas and results in the field, do explorations and fairly elementary proofs, and become aware of some recent questions. By presenting a wide range of topics, exercises, and examples, Goodman creates an interactive and enjoyable atmosphere in which to learn topology.
Benefits:
0. AN OVERVIEW.
1. INTRODUCTION TO POINT SET TOPOLOGY.
Open and Closed Sets. Continuous Functions. Some Topological Properties. A Brief Introduction to Dimension.
2. SURFACES.
Definition of a Surface. Connected Sum Construction. Plane Models of Surfaces. Orientability. Plane Models of Nonorientable Surfaces. Classification of Surfaces. Proof of the Classification Theorem for Surfaces.
3. THE EULER CHARACTERISTIC.
Cell Complexes and the Euler Characteristic. Triangulations. Genus. Regular Complexes. b-valent Complexes.
4. MAPS AND GRAPHS.
Maps and Map Coloring. The 5-color Theorem for S². Introduction to Graphs. Graphs in Surfaces. Imbedding the Complete Graphs, and Graph Coloring.
5. VECTOR FIELDS ON SURFACES.
Vector Fields in the Plane. Index of a Critical Point. Limit Sets in the Plane. A Local Description of a Critical Point. Vector Fields on Surfaces. Proof of the Poincaré-Hopf Index Theorem.
6. THE FUNDAMENTAL GROUP.
Path Homotopy and the Fundamental Group. The Fundamental Group of the Circle. Deformation Retracts. Further Calculations. Presentations of Groups. The Seifert-van Kampen Theorem and the Fundamental Groups of Surfaces. Proof of the Seifert-van Kampen Theorem.
7. INTRODUCTION TO KNOTS.
Knots--What they are and How to Draw Them. Prime Knots. Alternating Knots. Reidemeister Moves. Some Simple Knot Invariants. Surfaces with Boundary. Knots and Surfaces. Knot Polynomials. A Short Knot Table.
With a nice balance of mathematical precision and accessibility, this text provides a broad introduction to the field of topology. Author Sue Goodman piques student curiosity and interest without losing necessary rigor so that they can appreciate the beauty and fun of mathematics. The text demonstrates that mathematics is an active and ever-changing field with many problems still unsolved, and students will see how the various areas of mathematics-algebra, combinatorics, geometry, calculus, and differential equations-interact with topology. Students learn some of the major ideas and results in the field, do explorations and fairly elementary proofs, and become aware of some recent questions. By presenting a wide range of topics, exercises, and examples, Goodman creates an interactive and enjoyable atmosphere in which to learn topology.
Benefits:
Table of Contents
0. AN OVERVIEW.
1. INTRODUCTION TO POINT SET TOPOLOGY.
Open and Closed Sets. Continuous Functions. Some Topological Properties. A Brief Introduction to Dimension.
2. SURFACES.
Definition of a Surface. Connected Sum Construction. Plane Models of Surfaces. Orientability. Plane Models of Nonorientable Surfaces. Classification of Surfaces. Proof of the Classification Theorem for Surfaces.
3. THE EULER CHARACTERISTIC.
Cell Complexes and the Euler Characteristic. Triangulations. Genus. Regular Complexes. b-valent Complexes.
4. MAPS AND GRAPHS.
Maps and Map Coloring. The 5-color Theorem for S². Introduction to Graphs. Graphs in Surfaces. Imbedding the Complete Graphs, and Graph Coloring.
5. VECTOR FIELDS ON SURFACES.
Vector Fields in the Plane. Index of a Critical Point. Limit Sets in the Plane. A Local Description of a Critical Point. Vector Fields on Surfaces. Proof of the Poincaré-Hopf Index Theorem.
6. THE FUNDAMENTAL GROUP.
Path Homotopy and the Fundamental Group. The Fundamental Group of the Circle. Deformation Retracts. Further Calculations. Presentations of Groups. The Seifert-van Kampen Theorem and the Fundamental Groups of Surfaces. Proof of the Seifert-van Kampen Theorem.
7. INTRODUCTION TO KNOTS.
Knots--What they are and How to Draw Them. Prime Knots. Alternating Knots. Reidemeister Moves. Some Simple Knot Invariants. Surfaces with Boundary. Knots and Surfaces. Knot Polynomials. A Short Knot Table.