Introduction to Topology: Pure and Applied by Adams and Franzosa

Introduction to Topology: Pure and Applied, by Colin Adams and Robert Franzosa

• Preface
• 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. One-Point Compactifications
• 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