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'Im trying to solve the following problem:

What figure does one obtain from a Möbius band if one shrinks the boundary circle to a point?

I don't really quite understand the problem. What does it mean the boundary circle of Möbius band? Can someone explain me what is this question asking for? thanks in advance.

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If you make a cylinder with a strip of paper, there will be two edges, a top and bottom. If you make a Möbius band, these two edges will just be a single contiguous edge, which is what is meant by "boundary circle".

The easiest way to see what happens when you contract this circle is to draw a rectangle with arrows to indicate how two opposite edges will be glued together. Now squeeze the two sides that aren't being identified, until you have a circle with two marked points (though they are identified in the gluing), and two arrows on either side going in opposite directions.

So now it's clear: you get a disc, with its boundary glued along the antipodal map. In other words, you get the real projective plane.

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  • $\begingroup$ How do you squeeze sides that aren't being identified to get a circle? $\endgroup$ Apr 30 at 8:29

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