Con($\mathfrak{r}_{\mathsf{nwd}}<\mathfrak{irr}$)
Jonathan Cancino Manríquez
TL;DR
The paper proves the relative consistency of key inequalities linking reaping and irresolvability invariants for the rationals, notably $\mathfrak{r}_{\mathsf{nwd}}<\mathfrak{irr}$ and $\mathfrak{r}_{\mathbb{Q}}<\mathfrak{u}_{\mathbb{Q}}$, by a countable support iteration that destroys countable irresolvable spaces while preserving a dense $\mathbb{Q}$-reaping family. Central to the construction is a forcing $\mathbb{Q}(\mathscr{I})$ based on saturated ideals, equipped with the Sacks property and a preservation theorem ensuring selective rational filters survive the iteration. The work also analyzes the existence and nonexistence of rational $p$- and $q$-filters under various models and diamond principles, tying these combinatorial objects to topological irresolvability. Consequently, the authors establish a model with $\mathfrak{r}_{\mathsf{scattered}}=\mathfrak{r}_{\mathbb{Q}}=\omega_1<\omega_2=\mathfrak{irr}=\mathfrak{i}=2^{\omega}$ and demonstrate $\mathfrak{r}_{\mathbb{Q}}<\mathfrak{u}_{\mathbb{Q}}$, advancing the understanding of the interaction between Boolean‑algebra invariants and the topology of irresolvable spaces.
Abstract
We prove the consistency of the inequality $\mathfrak{r}_{\mathsf{nwd}}<\mathfrak{irr}$, which in turn implies the consistency of $\mathfrak{r}_\mathsf{nwd}<\mathfrak{i}$ and $\mathfrak{r}_{\mathsf{scatt}}<\mathfrak{irr}$. This answers one question from \cite{balzar_hrusak_hernandez} and one question from \cite{cancino_irresolvable_1}. We also prove the consistency of the inequality $\mathfrak{r}_\mathbb{Q}<\mathfrak{u}_\mathbb{Q}$.