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