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- Path_(topology) abstract "In mathematics, a path in a topological space X is a continuous function f from the unit interval I = [0,1] to Xf : I → X.The initial point of the path is f(0) and the terminal point is f(1). One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path. Note that a path is not just a subset of X which "looks like" a curve, it also includes a parameterization. For example, the maps f(x) = x and g(x) = x2 represent two different paths from 0 to 1 on the real line.A loop in a space X based at x ∈ X is a path from x to x. A loop may be equally well regarded as a map f : I → X with f(0) = f(1) or as a continuous map from the unit circle S1 to Xf : S1 → X.This is because S1 may be regarded as a quotient of I under the identification 0 ∼ 1. The set of all loops in X forms a space called the loop space of X.A topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into a set of path-connected components. The set of path-connected components of a space X is often denoted π0(X);.One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X is a topological space with basepoint x0, then a path in X is one whose initial point is x0. Likewise, a loop in X is one that is based at x0.".
- Path_(topology) thumbnail Path.svg?width=300.
- Path_(topology) wikiPageID "1189553".
- Path_(topology) wikiPageLength "6695".
- Path_(topology) wikiPageOutDegree "37".
- Path_(topology) wikiPageRevisionID "613452276".
- Path_(topology) wikiPageWikiLink Algebraic_topology.
- Path_(topology) wikiPageWikiLink Associative.
- Path_(topology) wikiPageWikiLink Associative_property.
- Path_(topology) wikiPageWikiLink Automorphism.
- Path_(topology) wikiPageWikiLink Automorphism_group.
- Path_(topology) wikiPageWikiLink Binary_operation.
- Path_(topology) wikiPageWikiLink Category:Homotopy_theory.
- Path_(topology) wikiPageWikiLink Category:Topology.
- Path_(topology) wikiPageWikiLink Category_(mathematics).
- Path_(topology) wikiPageWikiLink Category_theory.
- Path_(topology) wikiPageWikiLink Connected_space.
- Path_(topology) wikiPageWikiLink Continuous_function.
- Path_(topology) wikiPageWikiLink Continuous_function_(topology).
- Path_(topology) wikiPageWikiLink Coordinate_system.
- Path_(topology) wikiPageWikiLink Curve.
- Path_(topology) wikiPageWikiLink Endomorphism.
- Path_(topology) wikiPageWikiLink Equivalence_class.
- Path_(topology) wikiPageWikiLink Equivalence_relation.
- Path_(topology) wikiPageWikiLink Fundamental_group.
- Path_(topology) wikiPageWikiLink Fundamental_groupoid.
- Path_(topology) wikiPageWikiLink Group_(mathematics).
- Path_(topology) wikiPageWikiLink Groupoid.
- Path_(topology) wikiPageWikiLink Homotopy.
- Path_(topology) wikiPageWikiLink Homotopy_theory.
- Path_(topology) wikiPageWikiLink Isomorphism.
- Path_(topology) wikiPageWikiLink Loop_(topology).
- Path_(topology) wikiPageWikiLink Loop_space.
- Path_(topology) wikiPageWikiLink Mathematics.
- Path_(topology) wikiPageWikiLink Morphism.
- Path_(topology) wikiPageWikiLink Path-connected_component.
- Path_(topology) wikiPageWikiLink Path-connected_space.
- Path_(topology) wikiPageWikiLink Pointed_space.
- Path_(topology) wikiPageWikiLink Quotient_space_(topology).
- Path_(topology) wikiPageWikiLink Seifert–van_Kampen_theorem.
- Path_(topology) wikiPageWikiLink Topological_space.
- Path_(topology) wikiPageWikiLink Unit_circle.
- Path_(topology) wikiPageWikiLink Unit_interval.
- Path_(topology) wikiPageWikiLink Van_Kampens_Theorem.
- Path_(topology) wikiPageWikiLink File:Homotopy_between_two_paths.svg.
- Path_(topology) wikiPageWikiLink File:Path.svg.
- Path_(topology) wikiPageWikiLinkText "Path (topology)".
- Path_(topology) wikiPageWikiLinkText "Path (topology)#Path composition".
- Path_(topology) wikiPageWikiLinkText "Path".
- Path_(topology) wikiPageWikiLinkText "continuous path".
- Path_(topology) wikiPageWikiLinkText "loop".
- Path_(topology) wikiPageWikiLinkText "loops".
- Path_(topology) wikiPageWikiLinkText "path (topology)".
- Path_(topology) wikiPageWikiLinkText "path".
- Path_(topology) wikiPageWikiLinkText "paths".
- Path_(topology) hasPhotoCollection Path_(topology).
- Path_(topology) wikiPageUsesTemplate Template:Main.
- Path_(topology) subject Category:Homotopy_theory.
- Path_(topology) subject Category:Topology.
- Path_(topology) hypernym Function.
- Path_(topology) type ProgrammingLanguage.
- Path_(topology) comment "In mathematics, a path in a topological space X is a continuous function f from the unit interval I = [0,1] to Xf : I → X.The initial point of the path is f(0) and the terminal point is f(1). One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path. Note that a path is not just a subset of X which "looks like" a curve, it also includes a parameterization.".
- Path_(topology) label "Path (topology)".
- Path_(topology) sameAs Weg_(Mathematik).
- Path_(topology) sameAs Chemin_(topologie).
- Path_(topology) sameAs מסילה_(מתמטיקה).
- Path_(topology) sameAs Arco_(topologia).
- Path_(topology) sameAs 경로_(위상수학).
- Path_(topology) sameAs Pad_(topologie).
- Path_(topology) sameAs Droga_(topologia).
- Path_(topology) sameAs Caminho_(topologia).
- Path_(topology) sameAs m.04fv5q.
- Path_(topology) sameAs Путь_(топология).
- Path_(topology) sameAs Лінія.
- Path_(topology) sameAs Q1366002.
- Path_(topology) sameAs Q1366002.
- Path_(topology) sameAs 道路_(拓扑学).
- Path_(topology) wasDerivedFrom Path_(topology)?oldid=613452276.
- Path_(topology) depiction Path.svg.
- Path_(topology) isPrimaryTopicOf Path_(topology).