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If we have a rectangle with vertices labelled cyclically as $1,2,3,4$, and if we identify the directed edge $(1,2)$ with the directed edge $(4,3)$, then the space we get is homeomorphic to a cylinder.

Now, in $3$D-space, we twist the directed edge $(1,2)$ twice by $180^0$ (around its center) and then attach it to the directed edge $(4,3)$; one would say that this space is still homeomorphic to cylinder.

But, I could not understand whether we can pass from the second cylinder to the first without cutting it? If not, how do we justify it?

Maths Rahul
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    Homeomorphism is not bound to any "ambient space" such as 3D space. The notion you are probably thinking of (transforming a shape into another in 3D space without cutting) is isotopy. See also this question: https://math.stackexchange.com/questions/2383751/how-is-a-doubly-twisted-cylinder-different-from-a-cylinder – Naïm Camille Favier Aug 09 '24 at 10:48
  • I recommend taking a thick rubber band (or anything shaped like a transmission belt), holding it between your hands and then twisting the part held by your right hand by 180 degrees. – Ben Steffan Aug 09 '24 at 16:31

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