On the Unique Representation of Families of Sets

Let X and Y be uncountable Polish spaces. A ⊂ X×Y represents a family of sets C provided each set in C occurs as an x-section of A. We say A uniquely represents C provided each set in C occurs exactly once as an xsection of A. A is universal for C if A represents C and every x-section of A is in C. A is uniquely universal for C if it is universal and uniquely represents C. We show that there is a Borel set in X × R which uniquely represents the translates of Q if and only if there is a Σ 2 Vitali set. Assuming V = L there is a Borel set B ⊂ ω with all sections Fσ sets and all non-empty Kσ sets are uniquely represented by B. Assuming V = L there is a Borel set B ⊂ X × Y with all sections Kσ which uniquely represents the countable subsets of Y . There is an analytic set in X ×Y with all sections ∆ 2 which represents all the ∆ 0 2 subsets of Y , but no Borel set can uniquely represent the ∆ 2 sets. This last theorem is generalized to higher Borel classes.