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.
HINT: To show the first part of the hint, think about fractional parts of $x$ and $y$. Once you’ve done that, you know that $R^2/\!\sim$ is homeomorphic to $[0,1]^2/\!\sim$. Which points of $[0,1]^2$ are related? Can you see why $[0,1]^2/\!\sim$ is a torus?