How can I see inside the projective plane the Möbius band?
I need to know how the Möbius Band appears inside the projective plane.
I know it is easy using identifications and algebraic topology.
I want to use parametrization of both the inside and the outside manifolds now.
The Projective Line $P^1R$ is sitting inside the projective plane $$P^2R=\left\{\left[x:y:z\right]\right\}$$ as $$P^1R=\left\{z=0\right\}$$ It is well known that $P^1R$ is homeomorphic to a circle. Now look at (edited:) some $\epsilon$-neighborhood of $P^1R$ for some positive $\epsilon$. You can check that is a nontrivial interval bundle over $P^1R$, so ist is a Möbius band.