Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Spaces of countable free set number and PFA
HTML articles powered by AMS MathViewer

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

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.

References
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2020): 54A25, 03E35, 54D10
  • Retrieve articles in all journals with MSC (2020): 54A25, 03E35, 54D10
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