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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]

A subset A of a vector space X is called \alpha-lineable whenever A contains, except for the null vector, a subspace of dimension \alpha. If X has a topology, then A is \alpha-spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of (\alpha,\beta)-lineability/spaceability were introduced as an attempt to shed light on more subtle issues. In this paper, among other results, we prove some general criteria for the notion of (\alpha,\beta)-lineability/spaceability and, as applications, we extend recent results of different authors. [ABSTRACT FROM AUTHOR]