026 MOC Topology

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Topics

Metric spaces

• Metric Function
• Pseudometric Function
  • Every metric function is a pseudometric function
• Metric Space
  • Different metrics yield different metric spaces
• Pseudometric Space
  • All pseudometric spaces induce metric spaces
• Metric Topology
• Open Balls (Metric Spaces)
• Open Sets in Metric Spaces

Topological Spaces

• Topology
• Topological Space
  • Topological spaces are generalizations of metric space topologies
  • The real line is a topological space
• Topological Basis
  • The metric topology is generated by open balls
• Hausdorff Space
  • All metric spaces are Hausdorff spaces
• Interior (Topology)
• Closure (Topology)
• Boundary (Topology)
• Topologically Continuous Function
  • Continuous functions map close inputs to close outputs
• Homeomorphism
• Topological Invariant
• Compact Set
• Connected Space
• Path-Connected Space
• Simply-Connected Space
• Second-Countable Space
• Homotopy
  • Homotopies induce equivalence relations
• Homotopy Group
• Homology Group
• Cohomology Group

Topological Space Examples

• Sierpiński Space

Related

• 026a MOC Manifolds