New Hindman spaces

Filipów, Rafał; Kowitz, Krzysztof; Kwela, Adam; Tryba, Jacek

doi:10.1090/proc/15720

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.

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 $D$).
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.

References

Mathias Beiglböck and Henry Towsner, Transfinite approximation of Hindman’s theorem, Israel J. Math. 191 (2012), no. 1, 41–59. MR 2970862, DOI 10.1007/s11856-011-0195-1
Lorenzo Carlucci, “Weak yet strong” restrictions of Hindman’s finite sums theorem, Proc. Amer. Math. Soc. 146 (2018), no. 2, 819–829. MR 3731714, DOI 10.1090/proc/13856
Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
RafałFilipów, On Hindman spaces and the Bolzano-Weierstrass property, Topology Appl. 160 (2013), no. 15, 2003–2011. MR 3097000, DOI 10.1016/j.topol.2013.08.007
RafałFilipów, Nikodem Mrożek, Ireneusz Recław, and Piotr Szuca, $\scr {I}$-selection principles for sequences of functions, J. Math. Anal. Appl. 396 (2012), no. 2, 680–688. MR 2961261, DOI 10.1016/j.jmaa.2012.06.050
RafałFilipów and Jacek Tryba, Convergence in van der Waerden and Hindman spaces, Topology Appl. 178 (2014), 438–452. MR 3276758, DOI 10.1016/j.topol.2014.10.019
Jana Flašková, Hindman spaces and summable ultrafilters, Presented at the 22nd Summer Conference on Topology and its Applications, 2007.
Jana Flašková, Ideals and sequentially compact spaces, Topology Proc. 33 (2009), 107–121. MR 2471564
H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Analyse Math. 34 (1978), 61–85 (1979). MR 531271, DOI 10.1007/BF02790008
Neil Hindman, The existence of certain ultra-filters on $N$ and a conjecture of Graham and Rothschild, Proc. Amer. Math. Soc. 36 (1972), 341–346. MR 307926, DOI 10.1090/S0002-9939-1972-0307926-0
Neil Hindman, Finite sums from sequences within cells of a partition of $N$, J. Combinatorial Theory Ser. A 17 (1974), 1–11. MR 349574, DOI 10.1016/0097-3165(74)90023-5
Michael Hrušák, Combinatorics of filters and ideals, Set theory and its applications, Contemp. Math., vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 29–69. MR 2777744, DOI 10.1090/conm/533/10503
Albin L. Jones, A brief remark on van der Waerden spaces, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2457–2460. MR 2052425, DOI 10.1090/S0002-9939-04-07351-4
Miroslav Katětov, Products of filters, Comment. Math. Univ. Carolinae 9 (1968), 173–189. MR 250257
Menachem Kojman, Hindman spaces, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1597–1602. MR 1887003, DOI 10.1090/S0002-9939-01-06238-4
Menachem Kojman, van der Waerden spaces, Proc. Amer. Math. Soc. 130 (2002), no. 3, 631–635. MR 1866012, DOI 10.1090/S0002-9939-01-06116-0
Menachem Kojman and Saharon Shelah, van der Waerden spaces and Hindman spaces are not the same, Proc. Amer. Math. Soc. 131 (2003), no. 5, 1619–1622. MR 1950294, DOI 10.1090/S0002-9939-02-06916-2
A. Kwela and J. Tryba, Homogeneous ideals on countable sets, Acta Math. Hungar. 151 (2017), no. 1, 139–161. MR 3594409, DOI 10.1007/s10474-016-0669-z
S. Mrówka, On completely regular spaces, Fund. Math. 41 (1954), 105–106. MR 63650, DOI 10.4064/fm-41-1-105-106
B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927), 212–216.