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I'm strunggling to prove that when we collapse the boundary of a Möbius strip we obtain the RP²

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You can regard the Möbius strip as a square with, say, the vertical sides identified with a twist. Then collapsing the strip's boundary to a point means collapsing the two horizontal sides to a point, which we can decompose into collapsing each of the horizontal sides to a point and then identifying those two points.

We can perform these identifications in any order we like. If we first collapse the horizontal sides, we can regard the result as a hemisphere, with the horizontal sides collapsed to opposite points and the vertical sides forming a great circle. If we now identify opposite points, this takes care of the remaining two identifications: The vertical sides get identified with a twist, and the two points to which we collapsed the horizontal sides are also identified. The result is homeomorphic to $\mathbb R\mathbb P^2$.

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