Klein bottle is a manifold

Question

How do we prove Klein bottle is a 2-manifold?

Define $f(x,y)=(x,y+1)$ and $g(x,y)=(x+1,1-y)$, and $G$ is the group generated by $f$ and $g$.

I know Klein bottle can be defined by the quotient $q:\mathbb{R}^2\rightarrow\mathbb{R}^2/G$.

How do we find an open set $U$ for each point in the Klein bottle s.t. $U\approx\mathbb{R}^2$?

I know $q$ is a covering space map. Will that help?

Answer 1

$q$ being a covering space map is indeed very useful. It means that each point in the quotient has a neighborhood $U$ that is mapped by $q^{-1}$ to a discrete collection of open sets, each of them homeomorphic to $U$, and each of them an open subset of $\Bbb R^2$.

By possibly taking an even smaller neighborhood inside $U$, you can easily make this homeomorphic to $\Bbb R^2$.