Proper group action

Summary

A continuous action of a topological group $G$ on a space $X$ is proper when the map $(g, x) \mapsto (g \cdot x, x)$ is a proper map. Informally, group elements that move a compact set back onto itself form a compact set. Proper actions have Hausdorff orbit spaces, closed orbits and compact stabilizers. They are the right setting for quotients such as homogeneous spaces $G/H$ and for the action of discrete groups on manifolds.