Editing Homogeneous space (section)

{{Short description|Topological space in group theory}}
[[File:Torus.png|300px|thumb|A [[torus]]. The standard torus is homogeneous under its [[diffeomorphism]] and [[homeomorphism]] groups, and the [[flat torus]] is homogeneous under its diffeomorphism, homeomorphism, and [[isometry group]]s. ]]

In [[mathematics]], a '''homogeneous space''' is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the [[Group action (mathematics)|action]] of a [[Group (mathematics)|group]]. Homogeneous spaces occur in the theories of [[Lie group]]s, [[algebraic group]]s and [[topological group]]s. More precisely, a homogeneous space for a [[Group (mathematics)|group]] ''G'' is a [[Empty set|non-empty]] [[manifold]] or [[topological space]] ''X'' on which ''G'' [[Group action (mathematics)|acts]] [[Group action (mathematics)#Transitivity properties|transitively]].  The elements of ''G'' are called the '''symmetries''' of ''X''. A special case of this is when the group ''G'' in question is the [[automorphism group]] of the space ''X'' – here "automorphism group" can mean [[isometry group]], [[diffeomorphism group]], or [[homeomorphism group]]. In this case, ''X'' is homogeneous if intuitively ''X'' looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism ([[differential geometry]]), or homeomorphism ([[topology]]). Some authors insist that the action of ''G'' be [[effective group action|faithful]] (non-identity elements act non-trivially), although the present article does not. Thus there is a [[Group action (mathematics)|group action]] of ''G'' on ''X'' that can be thought of as preserving some "geometric structure" on ''X'', and making ''X'' into a single [[orbit (group theory)|''G''-orbit]].