Domain 05 · Mathematics Codex
Topology
The mathematics of continuous deformation — what properties of space survive when you stretch, bend, and twist without tearing.
Scope
Point-set topology, algebraic topology, differential topology, knot theory, manifold theory, homotopy theory, homology and cohomology.
A topologist famously cannot distinguish a coffee cup from a doughnut — both have one hole. This apparent absurdity conceals one of the deepest ideas in mathematics: the study of properties that are preserved under continuous deformation. Topology underlies modern physics (the classification of phases of matter), data science (topological data analysis), and the Poincaré Conjecture — the only Millennium Prize problem solved so far.
Entries in this domain
Build status: This domain is in active build. Live entries are marked below. All other entries are planned and pending — see build status for the full picture.
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Topological Space
Pending
A set equipped with a collection of open sets satisfying three axioms — the minimal structure needed to define continuity.
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Poincaré Conjecture
Pending
Every simply-connected, closed 3-manifold is homeomorphic to the 3-sphere — proved by Grigori Perelman in 2003.
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Knot
Pending
A closed curve in three-dimensional space that cannot be untangled to a circle without cutting — the subject of knot theory.