Topology Cheat Sheet

Topology studies properties of spaces that remain invariant under continuous deformations—often described as "rubber-sheet geometry" where objects can be stretched, bent, or twisted without tearing or gluing. It provides the foundational language for modern mathematics, enabling rigorous treatment of continuity, convergence, and limits in arbitrary spaces beyond the familiar Euclidean setting. A key insight: topological concepts generalize metric notions (like open balls and convergence) to settings where no distance function exists, relying instead on the structure of open sets to capture geometric and analytical phenomena. Modern applications include topological data analysis (TDA), which uses persistent homology to extract shape-based features from high-dimensional datasets.