Let $G$ be a topological group and $\Gamma$ be a lattice in $G$. Consider the action of $G$ on the homogeneous space $G/\Gamma$ by left translation.
Let $F$ be a closed subgroup of $G$ and $x\in G/\Gamma$. Is it always true that $Fx$ is also closed in $G/\Gamma$?
If necessary,one can assume $G$ is a Lie group (I am not sure if this is relevant though)
Best Answer
No. Let $ G=\Bbb{R}, \Gamma=\Bbb{Z}, F=\langle a\rangle$ where $a$ is irrational. Then the quotient space is a circle and the orbits are not closed.