Non-Hausdorff orbit space of a covering action
= Non-Hausdorff orbit space of a covering action
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.