Möbius strip with edge identified - constructing map? 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? 

I would construct a homeomorphism from $A/\sim$ to the Möbius strip where $A$ is the region on the sphere bounded by two antipodal lines of latitude (let's think of them as the Arctic and Antarctic circles) and $\sim$ is the equivalence relation identifying antipodal points. Now deforming the Arctic/Antarctic regions into the north and south poles, induces a homeomorphism from the Möbius strip with edge squeezed to a point and ${S^2/\sim}\ = P^2$.
