When is a quotient map to an orbit space a closed map?

Question

$\pi :X\to X/G $ where $G$ is a group acting on X and $X/G $ is the orbit space associated with this action. Does $\pi $ always take closed sets to closed sets? (or in other words is $\pi$ a closed map?)

Answer 1

Let $G=\mathbb{R}$ act on $X=\mathbb{R}^2$ by translating along the second coordinate. That is,$$t(x,y)=(x,y+t).$$ The quotient space is $\mathbb{R}$, and the quotient map is projection on the first coordinate.

Let $S=\{xy=1\}\subset\mathbb{R}^2$. Then $S$ is closed, but its image is not.