9. Topology

• 9.1. Topological Spaces
  • 9.1.1. Introduction
  • 9.1.2. Open Sets, Closed Sets, Clopen Sets
  • 9.1.3. Finite-Closed Topology
  • 9.1.4. Functions and topologies
  • 9.1.5. \(T_0\) and \(T_1\) spaces
  • 9.1.6. Countable-closed topology
  • 9.1.7. Unions and intersections
• 9.2. Euclidean Topology
  • 9.2.1. Basis
  • 9.2.2. Basis for a Given Topology
• 9.3. Limit Points
  • 9.3.1. Limit Points and Closure
  • 9.3.2. Neighborhoods
  • 9.3.3. Connectedness
• 9.4. Homeomorphisms
  • 9.4.1. Subspaces
  • 9.4.2. Homeomorphisms
  • 9.4.3. Non-Homeomorphic Spaces
• 9.5. Continuous Mappings
  • 9.5.1. Continuous Mappings
  • 9.5.2. Intermediate Value Theorem
• 9.6. Metric Spaces
  • 9.6.1. Metric Spaces
  • 9.6.2. Metric Spaces as Topological Spaces
  • 9.6.3. Convergence of Sequences
  • 9.6.4. Completeness
  • 9.6.5. Contraction mappings
  • 9.6.6. Baire spaces
• 9.7. Compactness
  • 9.7.1. Compactness
  • 9.7.2. Heine-Borel Theorem
• 9.8. Finite Products
  • 9.8.1. The product topology
• 9.9. Notes
  • 9.9.1. Online examples
  • 9.9.2. References