# 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.

## Classification of Manifolds

I'm not sure if the classification of manifolds belongs here. This way it's starting to look like Wikipedia with lots and lots of information. What does the house style say about it? --barto (talk) 06:32, 3 May 2017 (EDT)

Yeah it does a bit. Some of our early contributors, during the evolution of our now-established house style, got many of their presentational ideas from Wikipedia, and started to design pages in its image. We felt uncomfortable about just deleting stuff because it didn't fit, with a view to working our way up to such pages from the ground up. This is clearly a page we never got round to properly processing.

Moreover it requires a lot of highly nontrivial work to build the theory of holomorphic covering maps, so it's strange to talk about universal holomorphic convering maps when even holomorphic mappings aren't mentioned on this page. --barto (talk) 06:39, 3 May 2017 (EDT)

The seeds of the solution are there in your own words. We wait till we have built the theory of holomorphic covering maps, and then when all the groundwork is in place, we will then be able to structure the material around that, and reshape these pages appropriately. --prime mover (talk) 06:48, 3 May 2017 (EDT)