Equivalence of atlases on Riemann surface

Question

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!

Answer 1

One purpose (in a sense the main purpose) of the charts $(U_\alpha,\varphi_\alpha)$ in an atlas is to specify which functions $f:R\to\mathbb C$ should be called holomorphic; namely, one requires that $f\circ\varphi_\alpha^{-1}$ be holomorphic in the usual ($\mathbb C\to\mathbb C$) sense on $\varphi_\alpha(U_\alpha)$, for every $\alpha$. The compatibility condition about $\varphi_\beta\circ\varphi_\alpha^{-1}$ in the definition of "atlas" ensures that the different $\alpha$'s agree about what "holomorphic" should mean in $U_\alpha\cap U_\beta$, so that an atlas provides a single, global notion of "holomorphic" for functions $R\to\mathbb C$. Similarly, the definition of "equivalence" of two atlases is just what's needed to ensure that, if you use each of the two atlases to produce such a global notion of "holomorphic", then these two notions coincide.