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".
[Math] Why isn’t $\mathbb{RP}^2$ orientable
algebraic-topologymanifolds
Best Answer
For geometric understanding of the real projective plane, I prefer to think of it as gotten from a closed circular disk by identifying opposite points on the boundary. If you accept this, then a Möbius band subset is easily found: take any diameter of your original circle, and widen it to a strip.