Let $X$ be a connected, locally path connected, locally compact metric space and $G$ be a group of homeomorphisms of $X$. Let $\bar{G}$ be the closure of $G$ in Homeo$(X)$ endowed with the compact open topology. Let $\pi : X \rightarrow X/G$ and $\phi : X/G \rightarrow X/\bar{G}$ be the natural projections.
Define $\psi : X/\bar{G} \rightarrow X/G$ as follows - given $z\in X/\bar{G}$ choose a point $y \in X/G$ which satisfies $\phi (y) = z$ and set $\psi(z) =y$.
My question is, why is $\psi$ well-defined? I can't see why it is independent of the choice of $y$. Isn't it possible for two elements of $X$ to be in different $G$ - orbits but the same $\bar{G}$ - orbit?
Thank you!