If we paste the boundary of disk by antipodal point, we get $\mathbb{R}P^2$, which is a two dimensional manifold. However, if we paste a disk by antipodal point, then what do we get?
My attempt: Because $D^2=\coprod_{\lambda\in[0,1]}S^1$, then we get $D^2/\sim=\coprod_{\lambda\in[0,1]}RP^1$.
- Is this a two dimensional manifold?
- If so, it is homomorphic to which kind of surfaces?