General Topology - Logic in computer science student project

Topologies

• topology on a set
• topology generated by a base (a family of subsets closed under finite intersections)
• examples
  • trivial topology
  • discrete topology

Continuous maps

• definition of a continuous map (inverse image preserves opens)
• identity map is continuous
• constant map is continuous
• composition of continuous maps is continuous
• every map from space with discrete topology is continuous
• every map to space with indiscrete topology is continuous
• definition of a homeomorphism
• identity map is a homeomorphism
• compositions of homeomorphism is homeomorphism
• inverse of homeomorphism is homeomorphism

Properties of spaces

• Topological properties: T₀, T₁, T₂, regular

Cantor space ℂ

• Cantor space ℂ (the set ℕ → {⊤, ⊥} with product topology)
• ℂ is Kolmogorov