I have proven that the family of homeomorphisms $f_{n,m}(x, y) = (x, (−1)^n y) + (n, m)$ acts properly discontinuously on $\mathbb{R}^2$. Now I should deduce that there exists a covering map of Klein bottle $p: \mathbb{R}^2 \to K$ using this action.
So the problem to me is to identify $\mathbb{R}^2/G$ as the Klein bottle. Why is it the case?
My knowledge is that the Klein bottle is the result of identifying a $[0,1]^2$ square with the relation $aba^{-1}b$.
Some thoughts
With similar situations I thought it could be useful to use Dyck's theorem on the representation of groups which gives an epimorphism between representations but stablishing the isomorphism can be harder.
Restricting $p$ to the unit square is a surjective continuous map which is injective on the interior. So you can specify this map just by figuring out how the edges are identified. For each edge, which $f_{n,m}$ map it to another edge of the unit square? Does it reverse orientation or not?
I'm not sure what the best source is but I learned this in a class where our texts were Munkres and Hatcher.