Let $G \subseteq \rm{Homeo}(\mathbb{R}^3)$ be the group of all homeomorphisms of the form $f_{(n,m,k)}:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ where $n,m,k \in \mathbb{Z}$, and $f_{(n,m,k)}(x,y,z)=(x+n, y+m, z+ny+k)$.
Show that $G \backslash \mathbb{R}^3$ with the quotient topology from the standard topology on $\mathbb{R}^3$ is a compact Hausdorff space, and that each point in $G \backslash \mathbb{R}^3$ is contained in an open neighborhood that is homeomorphic to an open set in $\mathbb{R}^3$
I have been looking at group actions and quotient spaces, as well as properly discontinuous actions, but am having trouble putting this together.