2022/02/01 by Alan Dow, Dow, Alan, István Juhász +1
Economics, Econometrics and Finance · Mathematics · #03E04 #54A25 #54A35 #54D10 #Advanced Topology and Set Theory #Economic theories and models #FOS: Mathematics #General Topology (math.GN) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2202.00356
openalex publication_date 2022/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main result of this paper is that, under PFA, for every \em regular space X with F(X) = ω we have |X| ≤ w(X)ω; in particular, w(X) ≤ \mathfrakc implies |X| ≤ \mathfrakc. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X with F(X) = ω such that w(X) = \mathfrakc and |X| = 2^\mathfrakc. 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) = ω such that w(X) = \mathfrakc and |X| = 2^\mathfrakc. 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) = ω. Moreover, any non-empty open set in X also has size 2^\mathfrakc, and thus answers one of the main problems of \citeJShSSz by providing in ZFC a SAU space with no isolated points.