topology

What I do

• Analyze topological spaces and continuity
• Work with open and closed sets
• Study compactness and connectedness
• Apply homotopy and homology groups
• Analyze metric spaces
• Study fixed point theorems

When to use me

When analyzing spatial properties that are preserved under continuous deformations.

Key Concepts

• Topological Space: Set with collection of open sets satisfying union/finite intersection closure
• Continuous Function: f^{-1}(open) is open; preserves topological properties
• Homeomorphism: Bijective continuous map with continuous inverse
• Compactness: Every open cover has finite subcover; Heine-Borel in ℝⁿ
• Connectedness: Cannot be partitioned into two nonempty open sets
• Homotopy: Continuous deformation between maps; homotopy equivalence