Spaces of countable free set number and PFA

Abstract:

The main result of this paper is that, under PFA, for every regular space $X$ with $F(X) = \omega$ we have $|X| \le w(X)^\omega$; 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 $F(X) = \omega$ such that $w(X) = \mathfrak {c}$ and $|X| = 2^\mathfrak {c}$.

We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space $X$ with $F(X) = \omega$ such that $w(X) = \mathfrak {c}$ and $|X| = 2^\mathfrak {c}$. In fact, this space $X$ has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in $X$ intersect, which is much stronger than $F(X) = \omega$. Moreover, any non-empty open set in $X$ also has size $2^\mathfrak {c}$, and thus our example answers one of the main problems of both Juhász, Soukup, and Szentmiklóssy [Topology Appl. 213 (2016), pp. 8–23] and Juhász, Shelah, Soukup, and Szentmiklóssy [Topology Appl. 323 (2023), Paper No. 108288, 15 pp.] by providing in ZFC a SAU space with no isolated points.