Consider the group action $G=\mathbb{Z}$ on $X=\mathbb{R}^2\setminus\{(0,0)\}$ given by $n \cdot (x,y)=(2^nx,2^{-n}y)$. How to prove that $X/G$ isn't a Hausdorff space? I tried by searching 2 points in the quotient that are not "separated" by 2 open disjointed sets, but this seems to go nowhere. What can I do?
2026-04-06 05:16:51.1775452611
A quotient topological space that isn't Hausdorff
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This stumped me for a bit as well, but ... have you considered the orbits of $(0,1)$ and $(1,0)$?