I'm looking at the space $\mathbb{R^2}/G$ where $G = \mathbb{Z^2}$ acts by $(n,m)(x,y) = ((-1)^mx+m,y+n))$ and I'm trying to show that this is a smooth surface. I am having a couple of problems.
- To find local coordinates about a point $[x,y]$ I want to look at the pre-image of a small disc around this point in $\mathbb{R^2}$ which is then loads of discs scattered around the plane in the places described by the action. I'd like to just take one of these discs and say we obviously have locally homeomorphic to an open disc but I'm slightly worried about the choice in all the different representatives and does this then really give us a homeomorphism to an open disc?
- Secondly, I want to show this space is Hausdorff which is relatively easy to do just by looking at two points and considering different cases (whether points are on boundary of unit square or not) and then finding open sets. I was wondering though if there is a quick way to see that the quotient space is Hausdorff, perhaps because of some obvious property of the group action?
Thanks for any help!