DMS Topology Seminar

Speaker: Will Brian, University of North Carolina, Charlotte

Title: Large metric spaces and partitions of the real line into Borel sets

Abstract: I will sketch a proof that, assuming $0^dagger$ does not exist, if there is a partition of $R$ into $ℵ_ω$ Borel sets, then there is also a partition of $R$ into $ℵ_{ω+1}$ Borel sets. (And the same is true for any singular cardinal of countable cofinality in place of $ℵ_ω$.) This contrasts starkly with the situation for cardinals with uncountable cofinality and their successors, where the spectrum of possible sizes of partitions of R into Borel sets can (via forcing) be made completely arbitrary. The proof of this fact for $ℵ_ω$ uses the structure of a certain complete metric space of weight $ℵ_ω$, and the existence of a particular partition of that space into Polish spaces.