CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A topological groupoid $X_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} X_0$ is called proper if the continuous map
is a proper map.
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.
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.