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).