Topology is the study of structure preserved under continuous deformation. It ignores metric quantities in favor of invariants that survive stretching, bending, and twisting without tearing.
Continuity
Maps preserving neighborhoods define admissible deformation.
Homeomorphism
A bijective continuous map with continuous inverse. Two spaces related by such a map are topologically identical.
Invariants
Connectedness, compactness, genus, and homology classes remain unchanged under deformation.
Point-Set Topology
Foundations: open sets, convergence, continuity.
Algebraic Topology
Encodes global shape using algebraic invariants such as fundamental groups and homology.
Differential Topology
Studies smooth manifolds and differentiable maps.
Geometric Topology
Focuses on manifolds, knots, embeddings, and low dimensions.
Ricci flow evolves a Riemannian metric by diffusing curvature, analogous to a heat equation on geometry.
Singularities indicate topological obstruction. Surgery removes unstable regions while preserving global structure.
Given a smooth function \( f : M \to \mathbb{R} \), topology is revealed by the structure of level sets.
Topology changes only at critical values, governed by Morse theory.
Meromorphic functions are holomorphic except at isolated poles. These singularities encode global constraints.
Local singular behavior integrates into global invariants on Riemann surfaces.