New Hindman spaces
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- by Rafał Filipów, Krzysztof Kowitz, Adam Kwela and Jacek Tryba PDF
- Proc. Amer. Math. Soc. 150 (2022), 891-902 Request permission
Abstract:
We introduce a method that allows to turn topological questions about Hindman spaces into purely combinatorial questions about the Katětov order of ideals on $\mathbb {N}$. We also provide two applications of the method.
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We characterize $F_\sigma$ ideals $\mathcal {I}$ for which there is a Hindman space which is not an $\mathcal {I}$-space under the continuum hypothesis. This reduces a topological question of Albin L. Jones about consistency of existence of a Hindman space which is not van der Waerden to the question whether the ideal of all non AP-sets ($A\subseteq \mathbb {N}$ is an AP-set if it contains arithmetic progressions of arbitrary finite length) is not below the ideal of all non IP-sets ($A\subseteq \mathbb {N}$ is an IP-set if there exists an infinite set $D\subseteq \mathbb {N}$ such that $A$ contains all finite sums of distinct elements of $D$).
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Under the continuum hypothesis, we construct a Hindman space which is not an $\mathcal {I}_{1/n}$-space. This answers a question posed by Jana Flašková at the 22nd Summer Conference on Topology and its Applications.
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Additional Information
- Rafał Filipów
- Affiliation: Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland
- ORCID: 0000-0003-1568-8955
- Email: Rafal.Filipow@ug.edu.pl
- Krzysztof Kowitz
- Affiliation: Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland
- ORCID: 0000-0003-1878-6059
- Email: Krzysztof.Kowitz@phdstud.ug.edu.pl
- Adam Kwela
- Affiliation: Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland
- MR Author ID: 998822
- Email: Adam.Kwela@ug.edu.pl
- Jacek Tryba
- Affiliation: Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland
- MR Author ID: 1084963
- ORCID: 0000-0003-4799-8894
- Email: Jacek.Tryba@ug.edu.pl
- Received by editor(s): January 26, 2021
- Received by editor(s) in revised form: May 27, 2021
- Published electronically: December 1, 2021
- Communicated by: Heike Mildenberger
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 891-902
- MSC (2020): Primary 54A20, 05A17, 03E35; Secondary 03E50, 05C55, 11P99
- DOI: https://doi.org/10.1090/proc/15720
- MathSciNet review: 4356195