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Sep 30 at 19:31 comment added asv @Petrunin: Oops, you are right. Thank you. This is the final answer.
Sep 30 at 19:14 comment added Anton Petrunin Suppose your Moebius band is a rectangle $[0,1]\times[0,a]$ with glued $[0,1]\times\{0\}$ and $[0,1]\times\{a\}$. By sending $a\to 0$, you get a collapse to $[0,\tfrac12]$.
Sep 30 at 18:50 history asked asv CC BY-SA 4.0