My trek through MacLane and Birkhoff's Algebra has brought me to exercise 1.9.3, which defines the equivalence relation $E$ in $\Bbb R^2$ by $(x, y) E (x', y')$ iff $x-x'$ and $y-y'$ are both integers and asks me to prove that the quotient set $\Bbb R^2/E$ may be described as the set of points on a torus. Sadly, I can't fathom what this question is trying to express, since the quotient set appears to me to be a set of like grids of points that you could connect to make a bunch of $1\times 1$ squares if you wanted. I mean, a torus is three-dimensional last time I checked, so I don't know how you'd go about saying a partition of $\Bbb R^2$ is the same as a subset of $\Bbb R^3$. If someone could help me out in solving this problem, that'd be great. Thanks!