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.