I can picture in my mind why a square with opposites sides "glued" is topologically equivalent to a torus. I cannot however picture what happens when we define the torus as $\mathbb{R}^2 / \mathbb{Z}^2$. Can anyone explain to me intuituvely what happens in this case? I think the notation $\mathbb{R}^2 / \mathbb{Z}^2$ means that if the difference between two pairs in $\mathbb{R}^2$ is a pair of integers then they're equivalent. So in my mind instead of gluing opposites sides together what we do is like knitting the torus.
I guess I should construct some homeomorphism between the two spaces in order to prove that $\mathbb{R}^2 / \mathbb{Z}^2$ is a torus.