(Credit where due: this is a summary things from MathOverflow 55726 and Math Stack Exchange 1082834.)

Let *G* be a locally-compact group acting on a locally-compact Hausdorff topological space *X* (these additional conditions on *G* and *X* being certainly satisfied if e.g. *G *is finitely-generated and *X* is a proper metric space.)

We say the action is **properly discontinuous** if for any compact set , for at most finitely many . (cf. the action of on by translations, which is **discontinuous**: for any [sufficiently small] compact set *K*, for every ; here we allow finite overlap even at arbitrarily small resolution: this allows for finite stabilizers, parabolic elements, etc.)

In particular this implies that every point has an open neighborhood *U* s.t. for at most finitely many . (Take *U* to sit inside some compact set *K*.)

Note that the second condition is strictly weaker than the first: consider the -action on the punctured plane given by . This satisfies the second condition, but not the first—take *K* to be any closed annulus centered about the origin, for instance.

The main point of imposing these mildly technical conditions when we study “geometric” group actions is this:

This second condition is equivalent to **the quotient map being a covering map**. The first condition (the definition) serves to ensure that the map given by is proper, and **hence that the quotient X/G is Hausdorff.**