{"id":"topological-space","domain":"topology","type":"Concept","title":"Topological Space","slug":"topological-space","url":"/mathematics/topology/topological-space/","summary":"A set with a collection of open sets satisfying 3 axioms, giving a notion of nearness without distance. Foundation of topology.","formal_definition":"(X,τ): ∅,X∈τ; unions of τ-sets ∈τ; finite intersections of τ-sets ∈τ.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Munkres, J.R. (2000). Topology. 2nd ed. Prentice Hall."},{"tier":1,"citation":"Hausdorff, F. (1914). Grundzüge der Mengenlehre. Veit & Co."},{"tier":2,"citation":"Hatcher, A. (2002). Algebraic Topology. Cambridge University Press."},{"tier":3,"citation":"James, I.M. (ed.) (1999). History of Topology. North-Holland."}],"relationships":[{"type":"generalises","target":"limit","label":"Limit"},{"type":"key_result","target":"poincare-conjecture","label":"Poincaré Conjecture"},{"type":"historical_origin","target":"graph-theory","label":"Graph Theory"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.6"}