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Pseudointersection numbers, ideal slaloms, topological spaces, and cardinal inequalities

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Abstract

We investigate several ideal versions of the pseudointersection number \(\mathfrak {p}\), ideal slalom numbers, and associated topological spaces with the focus on selection principles. However, it turns out that well-known pseudointersection invariant \(\mathtt {cov}^*({\mathcal I})\) has a crucial influence on the studied notions. For an invariant \(\mathfrak {p}_\mathrm {K}({\mathcal J})\) introduced by Borodulin-Nadzieja and Farkas (Arch. Math. Logic 51:187–202, 2012), and an invariant \(\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J})\) introduced by Repický (Real Anal. Exchange 46:367–394, 2021), we have

$$\begin{aligned} \min \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathtt {cov}^*({\mathcal I})\}=\mathfrak {p},\qquad \min \{\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}\le \mathtt {cov}^*({\mathcal I}), \end{aligned}$$

respectively. In addition to the first inequality, for a slalom invariant \(\mathfrak {sl_e}({\mathcal I},{\mathcal J})\) introduced in  Šupina (J. Math. Anal. Appl. 434:477–491, 2016), we show that

$$\begin{aligned} \min \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathfrak {sl_e}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}=\mathfrak {p}. \end{aligned}$$

Finally, we obtain a consistency that ideal versions of the Fréchet–Urysohn property and the strictly Fréchet–Urysohn property are distinguished.

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Notes

  1. For more recent developments on traditional selection principles, see, e.g., [8, 9, 27].

  2. A sequence \(\langle V_n :\ n\in \omega \rangle \) of subsets of X is called an \(\omega \)-cover, if for every n, \(\ V_n\ne X\), and for every finite \(F\subseteq X\) there is an n such that \(F\subseteq V_n\). Usually, uncountable covers are considered as well. However, an open \(\omega \)-cover on a Tychonoff space X has a countable \(\omega \)-subcover if and only if every finite power \(X^n\) is Lindelöf [19].

  3. \(\gamma \)-cover is a \(\text {Fin}\)-\(\gamma \)-cover, and \(\varGamma =\text {Fin}\)-\(\varGamma \), see below for \({\mathcal I}\)-\(\gamma \)-cover.

  4. For a schema \(\text {U}_{fin}({\mathcal P},{\mathcal R})\), see [35, 37].

  5. Note that \(\text {Fin}\)-large cover does not coincide with standard notion of a large cover, therefore our \(\text {Fin}\)-\(\Lambda \) is not \(\Lambda \) known in the literature on selection principles.

  6. We have that \(\mathtt {cov}^*(\mathcal {I})\ne +\infty \) if and only if \({\mathcal I}\) is tall. This is appropriate for a range of inequalities in the paper. Note that the definition of \(\mathtt {cov}^*({\mathcal I})\) for not tall ideals in [20] is different.

  7. See Corollary 11.6(2) as well.

  8. By Mathias, Jalali–Naini and Talagrand Theorem, an ideal \(\mathcal {J}\) has the Baire property if and only if \(\mathcal {J}\) is meager, see e.g., [15].

  9. Elements of \({\mathcal P}\), \({\mathcal R}\) in the schema \(\left( {\begin{array}{c}{\mathcal P}\\ {\mathcal R}\end{array}}\right) \) are families of sets.

  10. or X has \({{\mathcal P}\brack {\mathcal R}}_\square \)

  11. The alternative formulation of the latter one: For every family \({\mathcal V}\) which forms a countable open \(\omega \)-cover of a topological space X there is a \({\mathcal J}\)-\(\gamma \)-cover \(\langle V_m :\ m\in \omega \rangle \) of X such that \(V_m\in {\mathcal V}\), and a set \(V_m\) may be repeated \(\square \)-many times in the enumeration.

  12. Decreasing with respect to the first parameter and increasing with respect to the second parameter.

  13. In case of discrete topological space, the implications may be replaced with the equivalences.

  14. Essentially proved by M. Repický [29]. A particular case appears in [35], and is essentially proved by A. Kwela [25].

  15. The last equality, \(\mathtt {non}\left( \left[ {\mathcal I}\text {-}\varGamma ,\varGamma \right] \right) =\mathtt {cov}^*({\mathcal I})\), is a particular case of the first one, since \(\mathfrak {p}_\square ({\mathcal I},\text {Fin})=\mathtt {cov}^*({\mathcal I})\). Moreover, this one was pointed out already in [35], and proved essentially in [29].

  16. Indeed, if \(\langle V_n :\ n\in \omega \rangle \in {\mathcal P}\) then we define a constant sequence with value \(\langle V_n :\ n\in \omega \rangle \), and apply principle \({\mathrm{S}}_1({\mathcal P},{\mathcal R})\).

  17. J. Gerlits and Zs. Nagy [19] considered arbitrary covers and families of functions instead of countable sequences.

  18. As already announced, it is shown in [35, Proposition 5.2(1)] as well.

  19. If we define \(*\) to be larger than each ideal, then \(\mathfrak {sl_e}(\triangle ,{\mathcal J})\) and \(\mathfrak {sl_t}(\triangle ,{\mathcal J})\) are decreasing with respect to the first coordinate, and increasing, decreasing, with respect to the second coordinate, respectively.

  20. \(\mathfrak {b}_{\mathcal J}\), \(\mathfrak {d}_{\mathcal J}\) is the minimal size of an unbounded, dominating family in \(({}^{\omega }\omega ,\le ^{\mathcal J})\), respectively. We write \(f\le ^{\mathcal J}g\) if \(\{n:\ f(n)>g(n)\}\in {\mathcal J}\).

  21. The first equality being proven already in [38].

  22. In case of discrete topological space, the implications may be replaced with the equivalences.

  23. Shown in [36] as well.

  24. The first equality in the first line has been shown in [36].

  25. It is a consequence of our Theorem 6.3 and Theorem 4.2(1) as well.

  26. The second line inequality in part (1) is proved in a proof of [35, Theorem 8.1(2)].

  27. I.e., we do not know whether the corresponding cardinal invariant is equal to other well-known cardinal invariant.

  28. i.e., if \(f\in \overline{A}\subseteq \mathrm {C}_p(X)\) then there is a countable set \(B\subseteq A\) such that \(f\in \overline{B}\).

  29. They are also called the countable strictly \({\mathcal J}\)-Fréchet–Urysohn property and the countable \({\mathcal J}\)-Fréchet–Urysohn property, respectively, see the end of the section.

  30. Excepts well-described cases with value \(+\infty \).

  31. The first equality being proven already in [38].

  32. J. Gerlits and Zs. Nagy [19] were interested in general versions of the properties without countability restriction. Hence, they have shown that the mentioned equivalences hold in a Tychonoff topological space considering arbitrary covers and families of functions.

  33. Note that corresponding assertions hold for an \({\mathrm{S}}_1({\mathcal I}\text {-}\varGamma ,{\mathcal J}\text {-}\varGamma )\)-space, see [10].

  34. The part about \(\mathfrak {sl_e}({\mathcal I},{\mathcal J})\) has been proven in [36]. A direct combinatorial proof may be found in [34].

  35. A direct proof about ideals \({\mathcal J}_1,{\mathcal J}_2\) is trivial.

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Acknowledgements

We would like to thank the participants of Košice set-theoretical seminar for valuable comments and suggestions. We strongly thank to anonymous referee for his suggestions and remarks, which substantially improved the presentation in the paper. We dedicate the paper to the memory of Lev Bukovský, the founder of Košice set-theoretical group, who passed away when I was preparing the second revision of the paper.

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Correspondence to Jaroslav Šupina.

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Šupina, J. Pseudointersection numbers, ideal slaloms, topological spaces, and cardinal inequalities. Arch. Math. Logic 62, 87–112 (2023). https://doi.org/10.1007/s00153-022-00832-8

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