I am trying to do the proof of Svarc-Milnor lemma:
"Let $G$ be a group acting by isometries on a proper geodesic metric space $(X,d)$, furthermore, suppose that this action is proper (for any $B \subset X$ compact the set $\{g \in G | g \cdot B \cap B \neq \varnothing \}$ is finite) and cocompact (X\G is compact). Then $G$ is finitely generated and for all $x\in X$ the map $$ \begin{array}{l}{G \longrightarrow X} \\ {g \longmapsto g \cdot x}\end{array} $$ is a quasi-isometry."
I have a question (that maybe is basic), becouse at some point the book that I am following says that the natural projection $\pi: X \longrightarrow G\backslash X $ associated with the action is an open map. I am not sure why this happens. Any idea will help. Thanks.
HINT:
For a surjection $p\colon X\to Y$, $X$ topological space, and $Y$ with the quotient topology, the map $p$ will be continuous. Now, when is the image $p(U)$ of a set $U$ open? Precisely when $p^{-1}(p(U))$ is an open subset of $X$. In the case $Y=G\backslash X$, we have $$p^{-1}(p(U))=\cup_{g\in G}(g\cdot U)$$ Now, what happens if $U$ is open?