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I'm trying to show that the Möbius strip with boundary circle identified to a point is homeomorphic to $P^2$ (real projective space). I get geometrically why this is so, but, how does one generally construct maps between these spaces to show that they're homeomorphic?

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    If you take as your definition that a Moebius strip is the total-space of a non-orientable $I$-bundle over $S^1$, and $\mathbb RP^2$ as $S^2$ modulo antipodal identification, the homeomorphism comes from observing that when you remove the "poles" (one point) from $\mathbb RP^2$ the resulting manifold is an $I$-bundle over the equatorial lines, then check it's non-orientable.2010-10-11

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