I've got a very vague question about Riemann surfaces, more like a meta-question:
One of the first things one defines is, obviously, an atlas on a Riemann surface $R$ ($R$ is a connected Hausdorff topological space), i.e. a family of open sets $U_{\alpha}$ together with homeomorphisms $\varphi_{\alpha}:U_{\alpha}\to \mathbb{C}$ such that $\bigcup_{\alpha}U_{\alpha}=R$ and such that the transition functions $\varphi_{\beta}\varphi_{\alpha}^{-1}$ are holomorphic.
Then two atlases $\mathcal{A}_1$ and $\mathcal{A}_2$ are called equivalent if their union is an atlas too.
Now my sort of meta-question: Why is this the right notion of equivalence? Why do we regard to atlases to be basically the same if their transition functions fit together well?
I get for example why homeomorphisms are the right thing for topological spaces, as then both their points as well as their open (i.e. basic) sets are in a one-to-one correspondence.
But for two atlases on a Riemann surface I'm not quite sure how I should think about it, yet.
Any help from you is very appreciated, hopefully you get my question. Cheers!