nLab
proper topological groupoid

Definition

A topological groupoid $X_1\stackrel{\stackrel{s}{\to }}{\underset{t}{\to }}{X}_0$ is called proper if the continuous map

$$(s,t):X_1\to X_0\times X_0$$(s,t):X_1 \to X_0\times X_0

is a proper map.

Properties

In particular the automorphism group of any object in a proper topological groupoid is a compact group. In this sense proper topological groupoids generalize compact groups.

Examples

A Lie groupoid is called a proper Lie groupoid if its underlying topological groupoid is proper.

An orbifold is a proper Lie groupoid which is also an étale groupoid.