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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. [ABSTRACT FROM AUTHOR]

Suppose that K is a compact Hausdorff space, S is a locally compact Hausdorff space and X is a Banach space with Schäffer constant \mathrm {S}(X). In this paper, we prove that if there is a map T from C(K) to C_{0}(S, X) satisfying \begin{equation*} \frac {1}{M} \|f-g\| \leq \|T(f)-T(g)\|\leq M \|f-g\|,\ \forall f, g \in C(K), \end{equation*} with 1\leq M^{2}<\mathrm {S}(X), then there exists a compact subset S_{0} of S and a continuous function \varphi from S_{0} onto K. This theorem on Lipschitz embeddings of C(K) into C_{0}(S, X) is the first nonlinear vector extension of the classical 1966 Holsztyński Theorem. Our result is optimal for many Banach spaces X including the spaces \ell _p^n, \ell _p and L_{p}([0,1]), 1 < p <\infty, n \geq 2, even when T is linear. The motivation to prove this result comes from the fact that it immediately yields a nontrivial lower bound for the C_{0}(S, X)-distortion of the class of all separable Banach spaces whenever S is a scattered space and \mathrm {S}(X)>1, namely \mathrm {S}(X) itself. [ABSTRACT FROM AUTHOR]