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

If yes, can you explain it to me?

Note: cup as in drinking container.

• What is a cup? ${}{}$ – Arctic Char Jun 26 at 15:28
• A drinking container – Sandrin Jun 26 at 15:29
• Then no in general – Arctic Char Jun 26 at 15:33
• Since a cup is 3-dimensional, it could only happen if $n=3$. Your quotient, when $n=3$ is a torus ("ring doughnut") which is topologically equivalent to a one-piece cup with handle since you could deform the body of the cup into a long tube connected at either end to the handle (the "hole" in the handle through which your fingers go becomes the hole in the torus). – postmortes Jun 27 at 5:11
• Depends on the $n$ and the cup. – tomasz Jun 27 at 11:05

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