First of all, I'm not sure if this is the right forum, but none of the forums mention topology in their description. But anyway, I'm taking a topology class, and the professor mentioned that the projective plane is obtained by identifying antipodal points on the sphere, ie, points diametrically opposite to each other. To get a better sense of what the space was like, he said it could also be obtained by considering just one half of the sphere and identifying opposite points on the boundary (equator). This seems reasonable, but I can't seem to come up with a rigorous proof that the two quotient spaces are homeomorphic. Can anyone help me out?