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1) I read that a Klein bottle is in fact a sphere with 2 disks removed and replaced by Möbius strips. I find it hard to imagine how this constructs a Klein bottle. Any ideas how I can convince myself in this statement?

2) Also, what if a single disk is removed from the sphere and replaced with a Möbius strip, does this object have some specific name? I think it's also called the projective plane. This is what my book says. Is this so? And why? From what I know the projective plane is something different. I don't see the connection or similarity to this object.

Alex Ravsky
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peter.petrov
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  • A klein bottle is homeomorphic to $\Bbb{RP}^2$ #$\Bbb{RP}^2$. And in 2), remove a disk from the sphere results a space that is homeomrphic to a disk and you can use this link to visualize how it is contructed: https://math.stackexchange.com/questions/2963241/how-to-see-the-real-projective-plane-is-a-möbius-band-glued-to-a-disk – Kevin.S Apr 16 '20 at 14:02
  • @Kevin.S Hm... My books says if you remove one disk and replace it with a Moebius strip you get the projective plane. If you replace two disks with Moebius strips you get a Klein bottle. That doesn't seem to match your statement, right? – peter.petrov Apr 16 '20 at 14:31
  • It matches my statement, maybe my sentences aren't clear which leads to confusion, sorry about that. Indeed 1) is the klein bottle that is homeomorphic to the connected sum of projective planes. For, the rest part of the sphere after the removal of a disk is again a disk if you then attach its boundary to the boundary of a mobius strip you get the projective plane, that's correct. – Kevin.S Apr 16 '20 at 14:39
  • @Kevin.S Oh... $#$ means connected sum. I got it. Then it all matches up, OK, thanks. – peter.petrov Apr 16 '20 at 15:54

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