If M is a topological manifold, a smooth structure A (or maximal atlas) on M is a set of smoothly compatible charts of M that is maximal in the sense that if we consider any chart that is not in A, then there is some chart in A with whom it is not smoothly compatible. Now, it is a fact that some topological manifolds admit no smooth structure and others admit many distinct ones. One could ask how many distinct smooth structures exists on a given manifold, but differential topologists choose to classify the smooth structures on a manifold up to diffeomorphism (meaning that two smooth structures A, A' are diffeomorphic if there exists a diffeomorphism F:(M,A)-->(M,A')). Could someone explain this choice to me? Because what is a smooth structure if not a way to make sense of differentiability of map on a topological space? So at first I thought that diffeomorphic smooth structures admit the same smooth maps (In the sense that for any smooth manifold N, a map G:(M,A)-->N is smooth if and only if G(M,A')-->N is smooth). But this is not the case, for the following reason: Lemma: Suppose that A, A' are two smooth differentiable structures on M that admit the same smooth maps. Then A = A'. Proof: To see this, consider (U,f) a chart on (M,A). Then f must be smooth as a map on (M,A'). This means that for any chart (V,g) in (M,A'), the coordinate representation of f, namely f o g^-1, is smooth. Inversely, given a chart (V,g) in (M,A'), g must be smooth on (M,A), so for any chart (U,f) in (M,A), the coordinate representation of g, namely g o f^-1, is smooth. So every pair of charts (U,f), (V,g) in A and A' respectively are smoothly compatible. This means that A=A'. QED If it were true that two diffeomorphic smooth structures admit the same smooth maps, then there would not exist distinct diffeomorphic smooth structures, which is absurd (see 2nd paragraph).