What is the definition of a diffeomorphism of an abstract smooth surface?

Question

I have the following definition for an abstract smooth surface:

I was also given a question asking me to prove that a specific function from the underlying topological set to itself is a diffeomorphism of the abstract smooth surface. This confuses me as for diffeomorphisms we require differentiability and the abstract smooth surface may not be differentiable in the usual sense. I suspect this has something to do with the transition maps but I am not sure.

Question: Does anybody know what is meant by a diffeomorphism of an abstract smooth surface?

Answer 1

Your definition looks like simplified definition of manifold (normlly we require $\Sigma$ to be Hausdorff). I assume the definition of atlas it the same.

Diffeomorephism between manifolds $M$ and $N$ is homeomorphism $f:M\to N$ such that for any chart $(U_i, \varphi_i)$ on $M$ and chart $(V_j, \psi_j)$ on $N$, composition $$ \psi_j \circ f \circ \varphi_i^{-1} $$ is (adequately many times) differentiable on its domain. Note that the domain is here some open subset of $\mathbb{R}^n$, same for image. This is function between subsets of euclidean spaces, for wchich we have standard notion of differentiability.