For juniors and seniors of various majors, taking a first course in topology. This book introduces topology as an important and fascinating mathematics discipline. Students learn first the basics of point-set topology, which is enhanced by the real-world application of these concepts to science, economics, and engineering as well as other areas of mathematics. The second half of the book focuses on topics like knots, robotics, and graphs. The text is written in an accessible way for a range of undergraduates to understand the usefulness and importance of the application of topology to other fields.
Colin Adams is the Thomas T. Read Professor of Mathematics at Williams College. He received his PhD from the University of Wisconsin-Madison in 1983. He is particularly interested in the mathematical theory of knots, their applications, and their connections with hyperbolic geometry. He is the author of The Knot Book, an elementary introduction to the mathematical theory of knots and co-author with Joel Hass and Abigail Thompson of How to Ace Calculus: The Streetwise Guide, and How to Ace the Rest of Calculus: the Streetwise Guide, humorous supplements to calculus. He has authored a variety of research articles on knot theory and hyperbolic 3-manifolds. A recipient of the Deborah and Franklin Tepper Haimo Distinguished Teaching Award from the Mathematical Association of America (MAA) in 1998, he was a Polya Lecturer for the MAA for 1998-2000, and is a Sigma Xi Distinguished Lecturer for 2000-2002. He is also the author of mathematical humor column called "Mathematically Bent" which appears in the Mathematical Intelligencer. Robert Franzosa is a professor of mathematics at the University of Maine. He received his Ph.D from the University of Wisconsin-Madison in 1984. He has published research articles on dynamical systems and applications of topology to geographic information systems. He has been actively involved in curriculum development and in education outreach activities throughout Maine. He is currently co-authoring a text, Algebraic Models in Our World, which is targeted for college-level general-education mathematics audiences. He was the recipient of the 2003 Presidential Outstanding Teaching Award at the University of Maine.
0. Introduction 0.1 What is Topology and How is it Applied? 0.2 A Glimpse at the History 0.3 Sets and Operations on Them 0.4 Euclidean Space 0.5 Relations 0.6 Functions 1. Topological Spaces 1.1 Open Sets and the Definition of a Topology 1.2 Basis for a Topology 1.3 Closed Sets 1.4 Examples of Topologies in Applications 2. Interior, Closure, and Boundary 2.1 Interior and Closure of Sets 2.2 Limit Points 2.3 The Boundary of a Set 2.4 An Application to Geographic Information Systems 3. Creating New Topological Spaces 3.1 The Subspace Topology 3.2 The Product Topology 3.3 The Quotient Topology 3.4 More Examples of Quotient Spaces 3.5 Configuration Spaces and Phase Spaces 4. Continuous Functions and Homeomorphisms 4.1 Continuity 4.2 Homeomorphisms 4.3 The Forward Kinematics Map in Robotics 5. Metric Spaces 5.1 Metrics 5.2 Metrics and Information 5.3 Properties of Metric Spaces 5.4 Metrizability 6. Connectedness 6.1 A First Approach to Connectedness 6.2 Distinguishing Topological Spaces Via Connectedness 6.3 The Intermediate Value Theorem 6.4 Path Connectedness 6.5 Automated Guided Vehicles 7. Compactness 7.1 Open Coverings and Compact Spaces 7.2 Compactness in Metric Spaces 7.3 The Extreme Value Theorem 7.4 Limit Point Compactness 7.5 The One-Point Compactification 8. Dynamical Systems and Chaos 8.1 Iterating Functions 8.2 Stability 8.3 Chaos 8.4 A Simple Population Model with Complicated Dynamics 8.5 Chaos Implies Sensitive Dependence on Initial Conditions 9. Homotopy and Degree Theory 9.1 Homotopy 9.2 Circle Functions, Degree, and Retractions 9.3 An Application to a Heartbeat Model 9.4 The Fundamental Theorem of Algebra 9.5 More on Distinguishing Topological Spaces 9.6 More on Degree 10. Fixed Point Theorems and Applications 10.1 The Brouwer Fixed Point Theorem 10.2 An Application to Economics 10.3 Kakutani's Fixed Point Theorem 10.4 Game Theory and the Nash Equilibrium 11. Embeddings 11.1 Some Embedding Results 11.2 The Jordan Curve Theorem 11.3 Digital Topology and Digital Image Processing 12. Knots 12.1 Isotopy and Knots 12.2 Reidemeister Moves and Linking Number 12.3 Polynomials of Knots 12.4 Applications to Biochemistry and Chemistry 13. Graphs and Topology 13.1 Graphs 13.2 Chemical Graph Theory 13.3 Graph Embeddings 13.4 Crossing Number and Thickness 14. Manifolds and Cosmology 14.1 Manifolds 14.2 Euler Characteristic and the Classification of Compact Surfaces 14.3 Three-Manifolds 14.4 The Geometry of the Universe 14.5 Determining which Manifold is the Universe
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