there are ternary relations S and T such that, if y is an element of A, then where S is in Γ and T is in the dual pointclass of Γ (that is, the complement of T is in Γ). Note here that we think of φn(x) as being ∞ whenever x∉A; thus the condition φn(x)≤φn(y), for y∈A, also implies x∈A. The definition does not imply that the collection of norms is in the intersection of Γ with the dual pointclass of Γ. This is because the three-way equivalence is conditional on y being an element of A. For y not in A, it might be the case that one or both of S(n,x,y) or T(n,x,y) fail to hold, even if x is in A (and therefore automatically φn(x)≤φn(y)=∞). Applications Scale property The scale property is a strengthening of the prewellordering property. For pointclasses of a certain form, it implies that relations in the given pointclass have a uniformization that is also in the pointclass. Periodicity Notes References Descriptive set theory