Definition talk:Riemann Surface

Changes
The previous definition wasn't quite correct - the point is that the transition maps should be holomorphic, and you cannot expect every open set (not even every open connected set) to be isomorphic to $$\C$$.

Also, it is common not to assume any countability axioms, since they can be shown to follow from the definition. I have given the standard definition now.

I'm not sure what the convention is for "complex manifold" - in fact, there are so many different definitions of manifolds anyway (do you assume metrizability; countable topology etc etc). It seems like maybe complex manifolds should have their own page anyway.

There are lots of terms where one should hunt for the correct links to terms and definitions. I didn't have time to do this right now; anyone who has time on their hands and would like to do so, feel free to take this on. Otherwise I'll come back to it at some point.