On the unique representation of families of sets

Abstract

Let X X and Y Y be uncountable Polish spaces. A ⊂ X × Y A \subset X\times Y represents a family of sets C \mathcal {C} provided each set in C \mathcal {C} occurs as an x x -section of A A . We say that A A uniquely represents C \mathcal {C} provided each set in C \mathcal {C} occurs exactly once as an x x -section of A A . A A is universal for C \mathcal {C} if every x x -section of A A is in C \mathcal {C} . A A is uniquely universal for C \mathcal {C} if it is universal and uniquely represents C \mathcal {C} . We show that there is a Borel set in X × R X\times R which uniquely represents the translates of Q \mathbb {Q} if and only if there is a Σ 2 1 \Sigma _2^1 Vitali set. Assuming V = L V = L there is a Borel set B ⊂ ω ω B \subset \omega ^\omega with all sections F σ F_\sigma sets and all non-empty K σ K_\sigma sets are uniquely represented by B B . Assuming V = L V =L there is a Borel set B ⊂ X × Y B \subset X\times Y with all sections K σ K_\sigma which uniquely represents the countable subsets of Y Y . There is an analytic set in X × Y X\times Y with all sections Δ 2 0 \Delta _2^0 which represents all the Δ 2 0 \Delta _2^0 subsets of Y Y , but no Borel set can uniquely represent the Δ 2 0 \Delta _2^0 sets. This last theorem is generalized to higher Borel classes.