内容

目次

0. Introduction 0.1 What is Topology and How is it Applied? 0.2 AGlimpse at the History 0.3 Sets and Operations on Them 0.4 Euclidean Space0.5 Relations 0.6 Functions 1. Topological Spaces 1.1 Open Sets and theDefinition of a Topology 1.2 Basis for a Topology 1.3 Closed Sets 1.4Examples of Topologies in Applications 2. Interior, Closure, and Boundary2.1 Interior and Closure of Sets 2.2 Limit Points 2.3 The Boundary of a Set2.4 An Application to Geographic Information Systems 3. Creating NewTopological Spaces 3.1 The Subspace Topology 3.2 The Product Topology 3.3The Quotient Topology 3.4 More Examples of Quotient Spaces 3.5Configuration Spaces and Phase Spaces 4. Continuous Functions andHomeomorphisms 4.1 Continuity 4.2 Homeomorphisms 4.3 The ForwardKinematics Map in Robotics 5. Metric Spaces 5.1 Metrics 5.2 Metrics andInformation 5.3 Properties of Metric Spaces 5.4 Metrizability 6.Connectedness 6.1 A First Approach to Connectedness 6.2 DistinguishingTopological Spaces Via Connectedness 6.3 The Intermediate Value Theorem 6.4Path Connectedness 6.5 Automated Guided Vehicles 7. Compactness 7.1 OpenCoverings and Compact Spaces 7.2 Compactness in Metric Spaces 7.3 TheExtreme Value Theorem 7.4 Limit Point Compactness 7.5 The One-PointCompactification 8. Dynamical Systems and Chaos 8.1 Iterating Functions8.2 Stability 8.3 Chaos 8.4 A Simple Population Model with ComplicatedDynamics 8.5 Chaos Implies Sensitive Dependence on Initial Conditions 9.Homotopy and Degree Theory 9.1 Homotopy 9.2 Circle Functions, Degree, andRetractions 9.3 An Application to a Heartbeat Model 9.4 The FundamentalTheorem of Algebra 9.5 More on Distinguishing Topological Spaces 9.6 Moreon Degree 10. Fixed Point Theorems and Applications 10.1 The Brouwer FixedPoint Theorem 10.2 An Application to Economics 10.3 Kakutani's Fixed PointTheorem 10.4 Game Theory and the Nash Equilibrium 11. Embeddings 11.1 SomeEmbedding Results 11.2 The Jordan Curve Theorem 11.3 Digital Topology andDigital Image Processing 12. Knots 12.1 Isotopy and Knots 12.2Reidemeister Moves and Linking Number 12.3 Polynomials of Knots 12.4Applications to Biochemistry and Chemistry 13. Graphs and Topology 13.1Graphs 13.2 Chemical Graph Theory 13.3 Graph Embeddings 13.4 CrossingNumber and Thickness 14. Manifolds and Cosmology 14.1 Manifolds 14.2 EulerCharacteristic and the Classification of Compact Surfaces 14.3Three-Manifolds 14.4 The Geometry of the Universe 14.5 Determining whichManifold is the Universe