Quotient map-torus

Question

Let $e_{1},e_{2}$ be a frame of $\mathbb{R}^{2}$. $G=(ne_{1}+me_{2} : n,m \in \mathbb{Z})$ be a subgroup acting on $\mathbb{R}^{2}$ by translation.Show that the quotient map $\pi :\mathbb{R}^{2}\to\mathbb{R}^{2}/G$ is locally bijective.

Answer 1

Let $x\in\mathbb{R}^2$ be arbitrary. Pick a neighbourhood $U$ of $x$ such that $|U\cap\mathbb{Z}^2|\le 1$. Then $\pi\left|_U\right.$ is bijective onto its image, since every $y\in U$ maps to a different equivalence class in $\mathbb{R}^2/G$.