I was trying to understand the following:
which I got from:
http://www.mit.edu/~9.520/fall14/slides/class14/class14_manifold.pdf
I was wondering, why the domain was:
$$ \alpha(U_{\alpha} \cap U_{\beta})$$
what confused me is the following:
$$\beta \circ \alpha^{-1}$$
means, apply beta to the result of alpha inverse i.e. $\beta ( \alpha^{-1}(x) )$. So I decided to follow the "function path". $\alpha^{-1}(x) \in U_{\alpha}$ and it goes into $\beta$. So its like $\beta( a' )$ where $a' \in U_{\alpha}$. But beta only maps things in $U_{\beta}$. Therefore, I see why we would need an intersection because $a'$ must also be in the domain of $\beta$. However, I don't see how come that the same as applying alpha to the intersection of both. If anything I would have thought that its applying beta to the intersection of both, not alpha.
The reason I think that is because we are doing $\beta ( \alpha^{-1}(x) )$. So $\alpha^{-1} \in U_{\alpha}$ and for it to be "processable" by beta it must be in beta also, so $\alpha^{-1} \in U_{\alpha} \cap U_{\beta}$ and THEN we apply beta. So I conclude its domain is:
$$\beta( U_{\alpha} \cap U_{\beta})$$
but I think thats wrong for some reason...