Topology

The study of properties of space preserved under continuous deformation (stretching, bending, twisting).

• core concept:: continuity defined through open sets, independent of distance
• homeomorphism identifies spaces that are topologically equivalent
• manifolds are spaces that locally resemble Euclidean space, the arena of relativity
• boundary, compactness, connectedness classify topological spaces
• algebraic topology uses groups (homotopy, homology) to distinguish spaces
• Euler characteristic is a topological invariant linking vertices, edges, and faces
• Foundation of differential equations, geometry, quantum mechanics, and graph theory
• Related: set theory, category theory, algebra, calculus