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I heard that you can obtain a real projective plane by gluing a disk to a Mobius band. But then I thought: if you cut a disk out of a Klein bottle (1 face, 0 edges) you'd get a shape with 1 face and 1 edges. This sounds a lot like a mobius band, but clearly isn't because that would mean reinserting the disk would yield a Klein bottle, not the aforementioned projective plane. What do you get instead?

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1 Answer

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A Klein bottle with a small disk removed is topologically the result of taking two Möbius bands and gluing an interval on the boundary of one to an interval on the boundary of the other. It's easy enough to make a paper model that an image seems superfluous.

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