Editing Partial function (section)

=== Charts and atlases for manifolds and fiber bundles ===

Charts in the [[Atlas (topology)|atlases]] which specify the structure of [[manifold]]s and [[fiber bundle]]s are partial functions. In the case of manifolds, the domain is the point set of the manifold. In the case of fiber bundles, the domain is the space of the fiber bundle. In these applications, the most important construction is the [[Atlas (topology)#Transition maps|transition map]], which is the composite of one chart with the inverse of another. The initial classification of manifolds and fiber bundles is largely expressed in terms of constraints on these transition maps.

The reason for the use of partial functions instead of functions is to permit general global topologies to be represented by stitching together local patches to describe the global structure. The "patches" are the domains where the charts are defined.