  The main result of this paper is that, under PFA, for every {\em 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 Hausdorff 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 {\em 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 answers
one of the main problems of \cite{JShSSz} by providing in ZFC a SAU space with
no isolated points.
