Filters, ideal independence and ideal Mrówka spaces (2304.04651v1)

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 $C\subseteq \hbox{spec}(\mathfrak{s}{mm})$. 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 Mr\'owka spaces for ideal independent families.