Editing Atlas (topology) (section)

==Transition maps==
{{ Annotated image | caption=Two charts on a manifold, and their respective '''transition map'''
| image=Two coordinate charts on a manifold.svg
| image-width = 250
| annotations =
{{Annotation|45|70|<math>M</math>}}
{{Annotation|67|54|<math>U_\alpha</math>}}
{{Annotation|187|66|<math>U_\beta</math>}}
{{Annotation|42|100|<math>\varphi_\alpha</math>}}
{{Annotation|183|117|<math>\varphi_\beta</math>}}
{{Annotation|87|109|<math>\tau_{\alpha,\beta}</math>}}
{{Annotation|90|145|<math>\tau_{\beta,\alpha}</math>}}
{{Annotation|55|183|<math>\mathbf R^n</math>}}
{{Annotation|145|183|<math>\mathbf R^n</math>}}
}}
A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the [[Inverse function|inverse]] of the other. This composition is not well-defined unless we restrict both charts to the [[Intersection (set theory)|intersection]] of their [[Domain of a function|domains]] of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)

To be more precise, suppose that <math>(U_{\alpha}, \varphi_{\alpha})</math> and <math>(U_{\beta}, \varphi_{\beta})</math> are two charts for a manifold ''M'' such that <math>U_{\alpha} \cap U_{\beta}</math> is [[Empty set|non-empty]].
The '''transition map''' <math> \tau_{\alpha,\beta}: \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\beta}(U_{\alpha} \cap U_{\beta})</math> is the map defined by
<math display="block">\tau_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.</math>

Note that since <math>\varphi_{\alpha}</math> and <math>\varphi_{\beta}</math> are both homeomorphisms, the transition map <math> \tau_{\alpha, \beta}</math> is also a homeomorphism.