Let R denote the real numbers.

Let S^2 denote the unit sphere in R^3

Let f: S^2 -> R^4 be defined by f(x,y,z)=(x^2-y^2,xy,yz,zx)

Can we prove that:

f determines a continuous map g: PR^2 -> R^4 where PR^2 is the real projective plane.

&

g is a homeomorphism onto a topological subspace of R^4