Spaces of countable free set number and PFA
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- by Alan Dow and István Juhász;
- Proc. Amer. Math. Soc. 151 (2023), 2253-2260
- DOI: https://doi.org/10.1090/proc/16248
- Published electronically: February 28, 2023
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Abstract:
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.
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Bibliographic Information
- Alan Dow
- Affiliation: Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223
- MR Author ID: 59480
- ORCID: 0000-0002-4643-1290
- Email: adow@uncc.edu
- István Juhász
- Affiliation: Alfréd Rényi Institute of Mathematics, Eötvös Loránd Research Network, Budapest, Hungary
- Email: juhasz@renyi.hu
- Received by editor(s): January 29, 2022
- Received by editor(s) in revised form: May 22, 2022, July 21, 2022, and August 26, 2022
- Published electronically: February 28, 2023
- Additional Notes: The second author was supported by NKFIH grant no. K129211.
- Communicated by: Vera Fischer
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 2253-2260
- MSC (2020): Primary 54A25, 03E35, 54D10
- DOI: https://doi.org/10.1090/proc/16248
- MathSciNet review: 4556215