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How to prove that $\mathbb{RP}^2$ isn't orientable? My book (do Carmo "Riemannian Geometry") gives a hint: "Show that it has a open subset diffeomorphic to the Möbius band", but I don't know even who is the "open subset".

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    Try following the hint: look for an open subset of the 2-sphere which becomes a Mobius band after you identify antipodal points.2012-04-18

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