Foundations of Topology
Geometry measures distances and angles; topology tracks properties unchanged by continuous deformation. Explore each principle from mathematical definition to physical evidence and limitations.
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Concept 01Topology versus geometryGeometry measures distances and angles; topology tracks properties unchanged by continuous deformation.Concept 02Open sets, continuity and compactnessTopology supplies a precise definition of continuity without first choosing a distance.Concept 03Connectedness and topological sectorsDisconnected regions of a configuration space cannot be connected by an allowed continuous path.Concept 04Homotopy and windingA loop can wrap a circle an integer number of times; smooth deformations cannot change that integer without breaking continuity.Concept 05Manifolds, spheres and toriA manifold looks locally like Euclidean space, even when its global topology is different.Concept 06Euler characteristic and genusCounting vertices, edges and faces reveals a global invariant of a closed orientable surface.Concept 07Fiber bundles and quantum eigenstatesA quantum eigenstate can be defined locally while a single smooth global phase convention may fail to exist.Concept 08Gauge transformations and observablesThe phase assigned to an eigenvector is conventional; measurable holonomy and curvature are not.