Quotient space of an action (orbit space)

Let $G$ act on a manifold $M$.

The quotient space of the action (also called the orbit space) is the set of orbits

equipped with the quotient topology for the canonical surjection

Concretely, a subset $U\subseteq M/G$ is open if and only if $\pi^{-1}(U)$ is open in $M$.

In general $M/G$ need not be a manifold (it may fail to be Hausdorff or locally Euclidean). Under the stronger hypothesis of a principal action , the orbit space becomes a smooth manifold (see quotient manifold ).

Examples

1. Rotations of the plane. For $SO(2)\curvearrowright \mathbb{R}^2$, the quotient $\mathbb{R}^2/SO(2)$ is naturally homeomorphic to $[0,\infty)$ via the radius function.
2. Translations by integers. For the action of $\mathbb{Z}$ on $\mathbb{R}$ by translations, the quotient $\mathbb{R}/\mathbb{Z}$ is (homeomorphic to) the circle $S^1$.
3. Conjugacy classes. For the conjugation action of a Lie group on itself, $G/G$ is the set of conjugacy classes; for many noncompact groups this quotient is not Hausdorff.