Topology homeomorphisms and quotient map question

In this R denotes the real numbers, and [.] denotes subscript

Let Homeo(X) be the group of homeomorphisms f:X->X where X is a topological space.

Let G be a subgroup of Homeo(X)

~ is an equivalence relation on G where x~y iff there exists g in G s.t g(x)=y

Q1:

Let p: X -> X/~ be the quotient map.

Prove that for every U open in X, p(U) is open in X/~

Q2:

Let X=R(^n)\{0}, n>1.

Let G be the subgroup of Homeo(X) composed of g[p](x)=px

Prove that X/~ is the real projective space PR^(n-1)

Also prove that the graph of the relation ~ is closed, and that PR^(n-1) is Hausdorff

Any help would be much appreciated,

KL