I know a differentiable manifold first of all is a set, then a topological space, then i give a smooth atlas.
Say the set is $R^2$, then i give the ordinary topology on $R^2$ and then i give an atlas with one chart so $A=\{(R^2,i_d)\}$, $i_d: R^2 \to \mathbb{R}^2$ the identity map.
I want your validation on something:
$R^2$, before i give the topology, is considered to be only a set, so i cant possibly say $(1,1)+(1,2)=(2,3)$, cause such an operation is not defined! The only thing i can say is that $R^2$ is an uncountable set!
On the other hand, the target set of the identity map, $\mathbb{R}^2$ is considered to be a vector space (or i can think it as a group the important is that it has more structure than that of a set) so i can do everything i use to do there, i.e. $(1,1)+(1,2)=(2,3)$!
(This is why i did the notational distinction $R^2$ and $\mathbb{R}^2$ for the case of a set and that of a field, respectively.)
Is all this correct?
Thank you