'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.


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.

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

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