The action is said to be proper if the map G × X → X × X defined by (g, x) ↦ (x, g ⋅ x) is proper. This means that given compact sets K, K ′ the set of g ∈ G such that g ⋅ K ∩ K ′ ≠ ∅ is compact. In particular, this is equivalent to proper discontinuity G is a discrete group.