Spaces of countable free set number and PFA

Dow, Alan and Juhász, István (2023) Spaces of countable free set number and PFA. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. ISSN 0002-9939

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Abstract

The main result of this paper is that, under PFA, for every regular space X with F (X) = ω we have |X| ≤ w(X)ω ; in particular, w(X) ≤ c implies |X| ≤ c. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X with F (X) = ω such that w(X) = c and |X| = 2c. 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) = c and |X| = 2c. 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) = ω. Moreover, any non-empty open set in X also has size 2c, and thus answers one of the main problems of [8] by providing in ZFC a SAU space with no isolated points.

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