Editing Orientability (section)

{{Short description|Possibility of a consistent definition of "clockwise" in a mathematical space}}
{{redirect2|Orientation (mathematics)|Orientation (space)|the orientation of a shape in a space|Orientation (geometry)|the orientation of a vector space|Orientation (vector space)|other uses|Orientation (disambiguation)}}

[[File:Torus.png|right|thumb|A [[torus]] is an orientable surface]]
[[File:Flipping in Möbius strip.gif|alt=Animation of a flat disk walking on the surface of a Möbius strip, flipping with each revolution.|thumb|The [[Möbius strip]] is a non-orientable surface. Note how the [[Disk (mathematics)|disk]] flips with every loop.]]
[[File:Steiner's Roman Surface.gif|right|thumb|The [[Roman surface]] is non-orientable.]]

In [[mathematics]], '''orientability''' is a property of some [[topological space]]s such as [[real vector space]]s, [[Euclidean space]]s, [[Surface (topology)|surface]]s, and more generally [[manifold]]s that allows a consistent definition of "clockwise" and "anticlockwise".<ref>{{Cite book|url=https://books.google.com/books?id=9MQ-AAAAIAAJ&q=oriented+manifold|title=Modern multidimensional calculus|last=Munroe|first=Marshall Evans|date=1963|publisher=Addison-Wesley |page=263}}</ref> A space is '''orientable''' if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an '''orientation''' of the space. Real vector spaces, Euclidean spaces, and [[sphere]]s are orientable. A space is '''non-orientable''' if "clockwise" is changed into "counterclockwise" after running through some [[loop (topology)|loop]]s in it, and coming back to the starting point. This means that a [[geometric shape]], such as [[File:Small pie.svg|20px]], that moves continuously along such a loop is changed into its own [[mirror image]] [[File:pie 2.svg|20px]]. A [[Möbius strip]] is an example of a non-orientable space.

Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of [[homology theory]], whereas for [[differentiable manifolds]] more structure is present, allowing a formulation in terms of [[differential form]]s.  A generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a [[fiber bundle]]) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.