I will link the following lecture notes, because it makes no sense to keep pasting from them.
When reading them, there are two things I do not understand. The author introduces smooth manifolds by defining charts (which in a first stage are just bijections with open images, Definition 2.2.1) and then shows how these charts define a topology on the manifolds (Proposition 2.2.5). He then shows that W.r.t. this topology the charts (Proposition 2.2.6).
But much later, he actually needs that charts are diffeomorphisms, or at least smooth – e.g., he defines on pp. 34 tangent maps $T_p f$ only for smooth maps $f$ and on the next page considers $T_p \psi $ for a chart $\psi$, so $\psi$ should be smooth as well?
So is a chart smooth, or a diffeomorphism? If not, the lecture notes contain errors, should the requirements of the definitions be relaxed?
Best Answer
In the definition, it is only required that overlap maps be smooth, as said in the answer of @AndresMejia.
But, here also is an answer to your question about whether a chart $\psi$ itself is smooth. Once smooth manifolds have been defined, and once smooth maps between smooth manifolds have been defined, it is then very easy exercise to prove that for each chart $\psi : U \to \mathbb R^n$ (where $U$ is some open subset of the manifold), the map $\psi$ is indeed smooth.