Why $\mathbb R^2/\mathbb Z^2$ and $\mathbb R/\mathbb Z\times \mathbb R/\mathbb Z$ are the same + $L^1(\mathbb R^2/\mathbb Z^2)$ ?

Question

Why $\mathbb R^2/ \mathbb Z^2$ is the Torus ? The Torus is indeed $\mathbb R/\mathbb Z\times \mathbb R/\mathbb Z$. But why $\mathbb R^2/\mathbb Z^2$ also gives the torus ? For me if we glue all points of $\mathbb Z^2$ together, then $\mathbb R^2/\mathbb Z^2$ should look as a sphere and not as a Torus ? Also, for me $L^1(\mathbb S^2)$ is the set of integrable periodic functions on $\mathbb R^2$. But in my course they wrote it as $L^1(\mathbb T^2)$ (the torus in $\mathbb R^3$). For me $L^1(\mathbb T^2)$ is the set of function $f$ s.t. $x\mapsto f(x,y )$ is $T_1$ periodic and $y\mapsto f(x,y)$ is $T_2$ periodic, but there is no reason to have $f$ periodic.