Q:

Is $\mathbb{R^n}/{\mathbb{Z}^n}$ homeomorphic to a cup?

If yes, can you explain it to me?

Note: cup as in drinking container.

A:

If you mean the following by a cup, then a cup is homeomorphic to $\mathbb{R}^2/\mathbb{Z}^2$, but not for $n \neq 2$.

