TITLE: How to show $\mathbb{R}^2/\mathbb{Z}^2$ is homeomorphic to $\mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}$
QUESTION [5 upvotes]: I know that $\mathbb{R}^2/\mathbb{Z}^2$ is homeomorphic to $S_1 \times S_1$ and $\mathbb{R}/\mathbb{Z}$ is homeomorphic to $S_1$ thus the product is homeomorphic to $S_1 \times S_1$. But I wonder if there's a way to show the quotient spaces are homeomorphic without having to refer to $S_1$.

REPLY [2 votes]: Consider two spaces $A,B$ with equivalence relations $R,S$. Then $R\times S$ is an equivalence relation, and there is a natural map $p:(A\times B)/(R\times S)\to A/R\times B/S$. This is always a continuous bijection, but might not be an homeomorphism. In your case since the spaces are compact Hausdorff, it is. One can weaken the hypothesis, however. I wouldn't know to what extent.