## Homeomorphism and real projective plane

Let S denote the unit sphere in ${R}^3$ given by $s = \{(x,y,z) \in {R}^3 : x^2 + y^2 + z^2 = 1\}$, and let $f:S \rightarrow {R}^4$ be defined by $f(x,y,z) = (x^2 - y^2, xy, yz, zx)$.

Prove that

a) f determines a continuous map $g: P \rightarrow {R}^4$ where P is the real projective plane.

b) g is a homeomorphism onto a topological subspace of ${R}^4$. (I can show g is injective, which requires a lot of work, so you don't have to do this bit).