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If yes, can you explain it to me?


Note: cup as in drinking container.

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    $\begingroup$ What is a cup? ${}{}$ $\endgroup$ – Arctic Char Jun 26 at 15:28
  • $\begingroup$ A drinking container $\endgroup$ – Sandrin Jun 26 at 15:29
  • $\begingroup$ Then no in general $\endgroup$ – Arctic Char Jun 26 at 15:33
  • $\begingroup$ 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). $\endgroup$ – postmortes Jun 27 at 5:11
  • $\begingroup$ Depends on the $n$ and the cup. $\endgroup$ – tomasz Jun 27 at 11:05
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If you mean the following by a cup, then a cup is homeomorphic to $\mathbb{R}^2/\mathbb{Z}^2$, but not for $n \neq 2$.

enter image description here

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  • $\begingroup$ This makes more sense, I probably wrote the exercise wrong $\endgroup$ – Sandrin Jun 26 at 15:32

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