# Collapsing the boundary of a Möbius Strip to a point

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

thanks

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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 $\mathbb R\mathbb P^2$.