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 ...

We study the class of first-countable Lindel\"of scattered spaces, or "FLS" spaces. While every $T_3$ FLS space is homeomorphic to a subspace $\mathbb Q$, $T_2$ spaces turns out be surprisingly rich. Our investigation these reveals close ties $Q$-sets, Lusin sets, and their relatives, cardinals $\mathfrak{b}$ $\mathfrak{d}$. Many natural questions about turn independent $\mathsf{ZFC}$. prove th...

We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese–Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s Conjecture for אω , we can find a counter-ex...

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in [4] and [3] as the basis for a representation theory of (Lie) 2-groups. Two important technical results are establishe...