On the unique representation of families of sets

Abstract:

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