The action is properly discontinuous if for every compact subset K ⊂ X there are only finitely many g ∈ G such that g ⋅ K ∩ K ≠ ∅. This is strictly stronger than wandering; for instance the action of Z on R ∖ {(0, 0)} given by n ⋅(x, y) = (2 x, 2 y) is wandering and free but not properly discontinuous.