Firstlet be a topological space. The sheaf of real continuous functions on is said to be a differential structure on if for any open set , any functions , and any , the superposition .
2002, Donal J. Hurley, Michael A. Vandyck, Topics in Differential Geometry: A New Approach Using D-Differentiation, Springer (with Praxis Publishing), page 29,
It is important toemphasise that, among thevariouschoices for and , some are intrinsic to the differential structure of the manifold . In other words, among all the operators of -differentiation, some arise from the differential structure of .[…]On the other hand, there exist operators of -differentiation that do not follow from the differential structure of .
Given a Hausdorff topological space with differential structures and (these being maximal smooth atlases), we say that and are equivalent if there is a diffeomorphism from with the first differential structure to with the second differential structure. Note that need not be the identity function.