An Ideal Zoo in the Baire Space

TL;DR

The paper investigates translations of Cantor-space σ-ideals into the Baire space by introducing parametrized families $f\mathcal{N}(h)$, $f\mathcal{S}(h)$, and $f\mathcal{E}(h)$ for $h\in \omega^\omega$ with $\limsup h(n)=\infty$. It establishes foundational properties (e.g., $f\mathcal{N}(h)$ as a $\sigma$-ideal with a $G_\delta$ base) and explores their relationships to classical ideals, along with the Fin-variants, leading to a rich array of chain/antichain phenomena and cardinal-invariant bounds. The analysis yields both inclusion and separation results among the fake families (e.g., $f\mathcal{E}(h)\subseteq f\mathcal{N}(h)$ under multiplicativity constraints on $h$ and various non-inclusion cases between $f\mathcal{N}(h)$ and $f\mathcal{S}(h)$), and demonstrates continuum-sized chains and antichains, deepening understanding of how null-like and small-like behaviour translates to the Baire space. Overall, the work clarifies the structure of these parametrized ideals, their interactions, and their implications for set-theoretic invariants in the Baire setting.

Abstract

In this paper, we study the translations into the Baire space of several well-known $σ$-ideals and families originally defined on the Cantor space, using their combinatorial characterizations. These include the ideals of null sets, small sets, those generated by closed measure-zero sets, and the meager sets, leading to their "fake" analogues in the Baire space. We also parametrize families related to null sets by functions from $ω^ω$. Several structural properties and relations between these families are investigated, including whether they form ideals, the existence of large chains and antichains, orthogonality, the $κ$-chain condition, and the determination of certain cardinal invariants.

Theorems & Definitions (70)

• Definition 1
• Definition 2
• Lemma 3
• Definition 4
• Definition 5
• Proposition 6
• proof
• Definition 7
• Proposition 8
• Proposition 9