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.