The main result of this paper is that, under PFA, for every regular space X X with F ( X ) = ω F(X) = \omega we have | X | ≤ w ( X ) ω |X| \le w(X)^\omega ; in particular, w ( X ) ≤ c w(X) \le \mathfrak {c} implies | X | ≤ c |X| \le \mathfrak {c} . This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X X with F ( X ) = ω F(X) = \omega such that w ( X ) = c w(X) = \mathfrak {c} and | X | = 2 c |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 X with F ( X ) = ω F(X) = \omega such that w ( X ) = c w(X) = \mathfrak {c} and | X | = 2 c |X| = 2^\mathfrak {c} . In fact, this space X X has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in X X intersect, which is much stronger than F ( X ) = ω F(X) = \omega . Moreover, any non-empty open set in X X also has size 2 c 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.