I have a problem with the following:
"Define a relation $\sim$ on $R^2$ by $(u,v) \sim (x,y)$ if and only if both $u-x$ and $v-y$ are integers. Show that for each point $(x,y) \in R^2$ there exists at least one point $(u,v) \in [0,1] \times [0,1]$ such that $(u,v) \sim (x,y)$ and deduce that $R^2/\sim$ is compact. ($\sim$ is an equivalence relation.)"
My guess is that it has something to do with homeomorphism and torus but could not move on. I would appreciate any help.