Covering space action

An action of $G$ on $X$ is a covering space action when every point has a neighbourhood $U$ disjoint from $gU$ for every nonidentity $g$. The quotient map is then locally a covering map, although without stronger properness assumptions its quotient need not be Hausdorff.

If $\Gamma$ is a discrete subgroup of a topological group $G$, then left translation by $\Gamma$ is a covering-space action. Choose an identity neighbourhood $U$ with $U{U}^{-1}\cap \Gamma =\{1\}$; the translates $\gamma U$ are pairwise disjoint, and the quotient map $G\to \Gamma \backslash G$ restricts to a homeomorphism on each translate.

On ${\mathbb{R}}^2\setminus \{0\}$, the powers of $T(x,y)=(2x,y/2)$ act freely with locally disjoint translates. Nevertheless $(2^{-n},1)\to (0,1)$ while $T^n(2^{-n},1)\to (1,0)$, so the two distinct limiting orbits cannot be separated in the quotient.

Lifting the hyperbolic-scaling action to the universal cover of ${\mathbb{C}}^{\ast }$ preserves the two inseparable limiting orbits. The covering surface is biholomorphic to $\mathbb{C}$, so a simply connected Riemann surface can still have a non-Hausdorff quotient by a covering space action of homeomorphisms.