An Ideal Zoo in the Baire Space

TL;DR

This paper explores how well-known ideals and families from the Cantor space translate into the Baire space, analyzing their structural properties and cardinal invariants using combinatorial methods.

Contribution

It introduces the concept of 'fake' analogues of classical ideals in the Baire space and investigates their structural and combinatorial properties.

Findings

Identified conditions under which these families form ideals.

Analyzed the existence of large chains and antichains.

Determined relationships between these families and cardinal invariants.

Abstract

In this paper, we study the translations into the Baire space of several well-known $\sigma$-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 $\omega^\omega$. Several structural properties and relations between these families are investigated, including whether they form ideals, the existence of large chains and antichains, orthogonality, the $\kappa$-chain condition, and the determination of certain cardinal invariants.