Thread: Topology question: continuous maps and homeomorphisms

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

for the first question we need only to verify that f(p)=f(-p), which is obvious. So if is the quotient map from to , q= (p)= (-p) is the image of p, g(q) is defined to be f(p). g is obviously continuous since g=f(\pi^{-1}), if we choose a sheet of the covering locally.
To see that g is a homeomorphism from M= to g(M), you need only to show that g is injective.