Partitions of Baire space into compact sets
Vera Fischer, Lukas Schembecker
TL;DR
The paper investigates partitions of uncountable Polish spaces, focusing on partitioning Baire space $\omega^\omega$ into compact sets and the invariant $\mathfrak{a}_T$. It introduces a c.c.c forcing that explicitly adds a partition into $\kappa$-many compact sets via maximal a.d.f.s. of finitely splitting trees, with diagonalization ensuring maximality and indestructibility under Sacks forcing. It then analyzes the spectrum $\mathrm{spec}(\mathfrak{a}_T)$ in product-Sacks models through a precise isomorphism-of-names argument, establishing cases where $\mathrm{spec}(\mathfrak{a}_T)=\{\aleph_1,\mathfrak{c}\}$ (and $\lambda$ when $\lambda^{\aleph_0}=\lambda$). Finally, it uses Shelah's ultrapower/template iterations to demonstrate models with $\mathfrak{d}<\mathfrak{a}=\mathfrak{a}_T=\mathfrak{c}$ relative to a measurable, yielding $\aleph_1<\mathfrak{d}<\mathfrak{a}=\mathfrak{a}_T$ and clarifying the relation between these cardinals in strong forcing frameworks.
Abstract
Under $\text{CH}$ we construct a partition of Baire space into compact sets, which is indestructible by countably supported iteration and product of Sacks forcing of any length, answering a question of Newelski. Further, we present an in-depth isomorphism-of-names argument for $\text{spec}(\mathfrak{a}_\text{T}) = \{\aleph_1, \mathfrak{c}\}$ in the product-Sacks model. Finally, we prove that Shelah's ultrapower model for the consistency of $\mathfrak{d} < \mathfrak{a}$ also satisfies $\mathfrak{a} = \mathfrak{a}_\text{T}$. Thus, consistently $\aleph_1 < \mathfrak{d} < \mathfrak{a} = \mathfrak{a}_\text{T}$ holds relative to a measurable.