Define a relation on $mathbb{R}^2$ by $(a, b)sim(c, d)$ if and

only if $(c-a, d- b) in mathbb{Z}$. Prove that $sim$ is an equivalence relation.

Identifying $mathbb{R}^2$ with the plane in the usual way, describe the most

natural transversal for $sim$ which you can find. What, if anything,

has this question to do with doughnuts?

I have already proven the equivalence relation, and know the solution of the second part, since it is provided, but I have problems with understanding it. I presume it’s asking about the equivalence classes?

Solution:

A natural transversal is $I times I$ where $I = {r | r in mathbb{R}, 0 leq r < 1}$. Join the top and bottom of $I$ to form a

tube, and bend the tube round to join the two circles as well. You

have the surface of a doughnut (a torus in mathematical language).