PARTITIONING THE REAL LINE INTO BOREL SETS

Abstract For which infinite cardinals $\kappa $ is there a partition of the real line ${\mathbb R}$ into precisely Borel sets? Work Lusin, Souslin, and Hausdorff shows that can be partitioned $\aleph _1$ sets. But other than this, we show spectrum possible sizes partitions sets fairly arbitrary. example, given any $A \subseteq \omega with $0,1 \in A$ , forcing extension in ${A = \{ n :\, \text {there } {{\mathbb R}} { }\aleph _n\text sets}\}}$ . We also look at corresponding question for closed that, like sets, set all uncountable such