Introduction to Topology: Pure and Applied

摘要

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