$\Psi$-Spaces and Semi-Proximality

TL;DR

This paper investigates semi-proximality in $ ext{Psi}$-spaces derived from almost disjoint families and ladder systems, providing characterizations and independence results within set-theoretic frameworks.

Contribution

It characterizes semi-proximality in $ ext{Psi}$-spaces for various families and shows independence results related to ladder systems and ZFC.

Findings

Semi-proximal almost disjoint families are nowhere MAD and anti-Luzin.

$ ext{Psi}$-spaces from uniformizable ladder systems are semi-proximal.

Existence of non-semi-proximal $ ext{Psi}$-spaces over ladder systems is independent of ZFC.

Abstract

We discuss the proximal game and semi-proximality in $\Psi$-spaces of almost disjoint families over an infinite countable set and $\Psi$-spaces of ladder systems on $\omega_1$. We show that a semi-proximal almost disjoint families must be nowhere MAD, anti-Luzin and characterize semi-proximality for a class of ${\mathbb R}$-embeddable almost disjoint families. We show that a $\Psi$-spaces defined from a uniformizable ladder system is semi-proximal and a $\Psi$-space defined on a $\clubsuit^*$ sequence is not semi-proximal. Thus the existence of non-semi-proximal $\Psi$-space over a ladder system is independent of ZFC.