Partitioning the Real Line into Borel Sets

Abstract

For which infinite cardinals is there a partition of the real line into precisely Borel sets? Work of Lusin, Souslin, and Hausdorff shows that can be partitioned into Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of into Borel sets can be fairly arbitrary. For example, given any with, there is a forcing extension in which. We also look at the corresponding question for partitions of into closed sets. We show that, like with partitions into Borel sets, the set of all uncountable such that there is a partition of into precisely closed sets can be fairly arbitrary.