Let X be a topological space. I can prove that the set Homeo(X) of homeomorphisms $\displaystyle f:X \rightarrow X$ 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 ~$\displaystyle _{G}$ y iff there exists $\displaystyle g \in G$ such that $\displaystyle g(x) = y$" is an equivalence relation.

Now the question: Let G and ~$\displaystyle _{G}$ be as above, and let $\displaystyle p: X \rightarrow X /$~$\displaystyle _{G}$ be the quotient map. Prove that for every U open in X, P(U) is open in X / ~ $\displaystyle _{G}$.

Let $\displaystyle X = {R}^n \ {0}, n\geq 2$, and let G be the subgroup of Homeo(X) composed of the maps $\displaystyle g(x) = cx$ where c is a constant. Prove that $\displaystyle x /$ ~$\displaystyle _{G}$ is the real projective space $\displaystyle P {R}^{n-1}$.