Openness in quotient space

Let X be a topological space. I can prove that the set Homeo(X) of homeomorphisms becomes a group when endowed with the binary operation of composition.

I can also show that if G is a subgroup of Homeo(X), then the relation "x ~ y iff there exists such that " is an equivalence relation.

Now the question: Let G and ~ be as above, and let ~ be the quotient map. Prove that for every U open in X, P(U) is open in X / ~ .

Let , and let G be the subgroup of Homeo(X) composed of the maps where c is a constant. Prove that ~ is the real projective space .