Filters, ideal independence and ideal Mrówka spaces

Abstract

A family $\mathcal{A}\subseteq [\omega {]}^{\omega }$ such that for all finite $\{X_i{\}}_{i\in n}\subseteq \mathcal{A}$ and $A\in \mathcal{A}\setminus \{X_i{\}}_{i\in n}$, the set $A\setminus {\bigcup }_{i\in n}{X}_i$ is infinite, is said to be ideal independent. We prove that an ideal independent family $\mathcal{A}$ is maximal if and only if $\mathcal{A}$ is $\mathcal{J}$-completely separable and maximal $\mathcal{J}$-almost disjoint for a particular ideal $\mathcal{J}$ on $\omega$. We show that $\mathfrak{u}\le {\mathfrak{s}}_{mm}$, where ${\mathfrak{s}}_{mm}$ is the minimal cardinality of maximal ideal independent family. This, in particular, establishes the independence of ${\mathfrak{s}}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality $\lambda$ for each $\lambda \in C$, thus establishing the consistency of $C\subseteq spec({\mathfrak{s}}_{mm})$. Assuming $\mathsf{CH}$, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, ${}^{\omega }\omega$-bounding, $p$-point preserving forcing notion and evaluate ${\mathfrak{s}}_{mm}$ in several well studied forcing extensions. We also study natural filters associated with ideal independence and introduce an analog of Mrówka spaces for ideal independent families.

18 Pages, subsumes arXiv:2206.14019