Spaces of countable free set number and PFA

Abstract

The main result of this paper is that, under PFA, for every {\em regular} space $X$ with $F(X)=\ddot{I}‰$ we have $|X|\le w(X{)}^{\ddot{I}}‰$; in particular, $w(X)\le \mathfrak{c}$ implies $|X|\le \mathfrak{c}$. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces $X$ with $F(X)=\ddot{I}‰$ such that $w(X)=\mathfrak{c}$ and $|X|=2^{\mathfrak{c}}$. We also show that regularity cannot be weakened to Hausdorff in this result because we can find in ZFC a Hausdorff space $X$ with $F(X)=\ddot{I}‰$ such that $w(X)=\mathfrak{c}$ and $|X|=2^{\mathfrak{c}}$. In fact, this space $X$ has the {\em strongly anti-Urysohn} (SAU) property that any two infinite closed sets in $X$ intersect, which is much stronger than $F(X)=\ddot{I}‰$. Moreover, any non-empty open set in $X$ also has size $2^{\mathfrak{c}}$, and thus answers one of the main problems of \cite{JShSSz} by providing in ZFC a SAU space with no isolated points.