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 $\mathfrak {u}\leq \mathfrak {s}_{mm}$, 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.