$\hat{I}¨$-Spaces and Semi-Proximality

Abstract

We discuss the proximal game and semi-proximality in $\hat{I}¨$-spaces of almost disjoint families over an infinite countable set and $\hat{I}¨$-spaces of ladder systems on $\ddot{I}{‰}_1$. 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 $\hat{I}¨$-spaces defined from a uniformizable ladder system is semi-proximal and a $\hat{I}¨$-space defined on a ${♣}^{\ast }$ sequence is not semi-proximal. Thus the existence of non-semi-proximal $\hat{I}¨$-space over a ladder system is independent of ZFC.

18 pages