Partitioning the real line into Borel sets

TL;DR

This paper investigates the possible sizes of partitions of the real line into Borel or closed sets, showing that the spectrum of such sizes can be highly flexible and depends on set-theoretic assumptions.

Contribution

It demonstrates that the set of possible partition sizes into Borel or closed sets can be arbitrarily complex, extending known results about partitions into Borel sets.

Findings

Existence of partitions into  Borel sets (Hausdorff)

Arbitrary spectrum of partition sizes into Borel sets under forcing extensions

Similar arbitrary spectrum results for partitions into closed sets

Abstract

For which infinite cardinals $\kappa$ is there a partition of the real line $\mathbb R$ into precisely $\kappa$ Borel sets? Hausdorff famously proved that there is a partition of $\mathbb R$ into $\aleph_1$ Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of $\mathbb R$ into Borel sets can be fairly arbitrary. For example, given any $A \subseteq \omega$ with $0,1 \in A$, there is a forcing extension in which $A = \{ n :\, \text{there is a partition of }\mathbb R\text{ into }\aleph_n\text{ Borel sets}\}$. We also look at the corresponding question for partitions of $\mathbb R$ into closed sets. We show that, like with partitions into Borel sets, the set of all uncountable $\kappa$ such that there is a partition of $\mathbb R$ into precisely $\kappa$ closed sets can be fairly arbitrary.