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YET ANOTHER IDEAL VERSION OF THE BOUNDING NUMBER

Part of: Set theory

Published online by Cambridge University Press:  13 September 2021

RAFAŁ FILIPÓW Open the ORCID record for RAFAŁ FILIPÓW [Opens in a new window]  and
ADAM KWELA Open the ORCID record for ADAM KWELA [Opens in a new window]
RAFAŁ FILIPÓW
Affiliation:
INSTITUTE OF MATHEMATICS, FACULTY OF MATHEMATICS PHYSICS AND INFORMATICS, UNIVERSITY OF GDAŃSK UL. WITA STWOSZA 57, GDAŃSK 80-308, POLAND E-mail: Rafal.Filipow@ug.edu.pl URL: http://mat.ug.edu.pl/~rfilipow E-mail: Adam.Kwela@ug.edu.pl URL: http://kwela.strony.ug.edu.pl/
ADAM KWELA
Affiliation:
INSTITUTE OF MATHEMATICS, FACULTY OF MATHEMATICS PHYSICS AND INFORMATICS, UNIVERSITY OF GDAŃSK UL. WITA STWOSZA 57, GDAŃSK 80-308, POLAND E-mail: Rafal.Filipow@ug.edu.pl URL: http://mat.ug.edu.pl/~rfilipow E-mail: Adam.Kwela@ug.edu.pl URL: http://kwela.strony.ug.edu.pl/
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Abstract

Let $\mathcal {I}$ be an ideal on $\omega $. For $f,\,g\in \omega ^{\omega }$ we write $f \leq _{\mathcal {I}} g$ if $f(n) \leq g(n)$ for all $n\in \omega \setminus A$ with some $A\in \mathcal {I}$. Moreover, we denote $\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$ (in particular, $\mathcal {D}_{\mathrm {Fin}}$ denotes the family of all finite-to-one functions).

We examine cardinal numbers $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$ and $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}}\times \mathcal {D}_{\mathrm {Fin}}))$ describing the smallest sizes of unbounded from below with respect to the order $\leq _{\mathcal {I}}$ sets in $\mathcal {D}_{\mathrm {Fin}}$ and $\mathcal {D}_{\mathcal {I}}$, respectively. For a maximal ideal $\mathcal {I}$, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers.

We show that $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}} \times \mathcal {D}_{\mathrm {Fin}})) =\mathfrak {b}$ for all ideals $\mathcal {I}$ with the Baire property and that $\aleph _1 \leq \mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) \leq \mathfrak {b}$ for all coanalytic weak P-ideals (this class contains all $\bf {\Pi ^0_4}$ ideals). What is more, we give examples of Borel (even $\bf {\Sigma ^0_2}$) ideals $\mathcal {I}$ with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\mathfrak {b}$ as well as with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) =\aleph _1$.

We also study cardinals $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {J}} \times \mathcal {D}_{\mathcal {K}}))$ describing the smallest sizes of sets in $\mathcal {D}_{\mathcal {K}}$ not bounded from below with respect to the preorder $\leq _{\mathcal {I}}$ by any member of $\mathcal {D}_{\mathcal {J}}\!$. Our research is partially motivated by the study of ideal-QN-spaces: those cardinals describe the smallest size of a space which is not ideal-QN.

Keywords

idealfilterultrafilterBorel idealP-idealweak P-idealbounding numberdominating numberquasi-normal convergencequasi-normal spaceQN-spacaadditivity of an ideal

MSC classification

Primary: 03E05: Other combinatorial set theory
Secondary: 03E15: Descriptive set theory 03E17: Cardinal characteristics of the continuum 03E35: Consistency and independence results

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Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 3 , September 2022 , pp. 1065 - 1092
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Bartoszyński, T. and Judah, H., Set Theory: On the Structure of the Real Line. Mathematical Reviews, A K Peters, Ltd., Wellesley, 1995.Google Scholar
Blass, A., Combinatorial cardinal characteristics of the continuum, Handbook of Set Theory (M. Foreman and A. Kanamori, editors), Springer, Dordrecht, 2010, pp. 395–489.CrossRefGoogle Scholar
Brendle, J. and Mejía, D. A., Rothberger gaps in fragmented ideals. Fundamenta Mathematicae, vol. 227 (2014), no. 1, 35–68.CrossRefGoogle Scholar
Brendle, J. and Shelah, S., Ultrafilters on $\omega$ —their ideals and their cardinal characteristics. Transactions of the American Mathematical Society, vol. 351 (1999), no. 7, 2643–2674.Google Scholar
Bukovský, L., On ${wQ}{N}_{\ast }$ and ${wQ}{N}^{\ast }$ spaces. Topology and its Applications, vol. 156 (2008), no. 1, 24.CrossRefGoogle Scholar
Bukovský, L., The structure of the real line, Instytut Matematyczny Polskiej Akademii Nauk [Mathematics Institute of the Polish Academy of Sciences], Birkhäuser/Springer Basel AG, Basel, 2011.Google Scholar
Bukovský, L., Das, P., and Šupina, J., Ideal quasi-normal convergence and related notions. Colloquium Mathematicum, vol. 146 (2017), no. 2, 265–281.CrossRefGoogle Scholar
Bukovský, L. and Haleš, J., ${QN}$ -spaces, ${wQN}$ -spaces and covering properties. Topology and its Applications, vol. 154 (2007), no. 4, 848–858.CrossRefGoogle Scholar
Bukovský, L., RecŁaw, I., and Repický, M., Spaces not distinguishing pointwise and quasinormal convergence of real functions. Topology and its Applications, vol. 41 (1991), no. 1–2, 25–40.CrossRefGoogle Scholar
Calbrix, J., Filtres boréliens sur l’ensemble des entiers et espaces des applications continues. Revue Roumaine de Mathématiques Pures et Appliquées, vol. 33 (1988), no. 8, 655–661.Google Scholar
Canjar, R. M., Model-theoretic properties of countable ultraproducts without the continuum hypothesis , Ph.D. thesis, University of Michigan, ProQuest LLC, Ann Arbor, MI, 1982.Google Scholar
Canjar, R. M., Countable ultraproducts without CH*. Annals of Pure and Applied Logic, vol. 37 (1988), no. 1, 1–79.CrossRefGoogle Scholar
Canjar, R. M., Cofinalities of countable ultraproducts: the existence theorem, Notre Dame Journal of Formal Logic, vol. 30 (1989), no. 4, 539–542.CrossRefGoogle Scholar
Das, P. and Chandra, D., Spaces not distinguishing pointwise and $\mathcal{I}$ -quasinormal convergence. Commentationes Mathematicae Universitatis Carolinae, vol. 54 (2013), no. 1, 83–96.Google Scholar
Debs, G. and Raymond, J. S., Filter descriptive classes of Borel functions. Fundamenta Mathematicae, vol. 204 (2009), no. 3, 189–213.CrossRefGoogle Scholar
Farah, I., Analytic quotients: theory of liftings for quotients over analytic ideals on the integers. Memoirs of the American Mathematical Society, vol. 148 (2000), no. 702, xvi+177.CrossRefGoogle Scholar
Farkas, B. and Soukup, L., More on cardinal invariants of analytic $P$ -ideals. Commentationes Mathematicae Universitatis Carolinae, vol. 50 (2009), no. 2, 281–295.Google Scholar
Filipów, R. and Staniszewski, M., Pointwise versus equal (quasi-normal) convergence via ideals. Journal of Mathematical Analysis and Applications, vol. 422 (2015), no. 2, 995–1006.CrossRefGoogle Scholar
Filipów, R. and Szuca, P., Three kinds of convergence and the associated $\mathcal{I}$ -Baire classes. Journal of Mathematical Analysis and Applications, vol. 391 (2012), no. 1, 1–9.CrossRefGoogle Scholar
Hernández-Hernández, F. and Hrušák, M., Cardinal invariants of analytic $P$ -ideals. Canadian Journal of Mathematics, vol. 59 (2007), no. 3, 575–595.CrossRefGoogle Scholar
Hrušák, M., Combinatorics of filters and ideals. Set Theory and its Applications, Contemporary Mathematics, 533, American Mathematical Society, Providence, 2011, pp. 29–69.CrossRefGoogle Scholar
Kwela, A., Ideal weak QN-spaces. Topology and its Applications, vol. 240 (2018), 98–115.CrossRefGoogle Scholar
Kwela, A. and RecŁaw, I., Ranks of $\mathcal{F}$ -limits of filter sequences. Journal of Mathematical Analysis and Applications, vol. 398 (2013), no. 2, 872–878.CrossRefGoogle Scholar
Kwela, A. and Sabok, M., Topological representations. Journal of Mathematical Analysis and Applications. vol. 422 (2015), no. 2, 1434–1446.CrossRefGoogle Scholar
Kwela, A. and Staniszewski, M., Ideal equal Baire classes. Journal of Mathematical Analysis and Applications, vol. 451 (2017), no. 2, 1133–1153.CrossRefGoogle Scholar
Laczkovich, M. and RecŁaw, I., Ideal limits of sequences of continuous functions. Fundamenta Mathematicae, vol. 203 (2009), no. 1, 39–46.CrossRefGoogle Scholar
Laflamme, C., Filter games and combinatorial properties of strategies, Set Theory (Boise, ID, 1992–1994), Contemporary Mathematics, 192, American Mathematical Society, Providence, 1996, pp. 51–67.Google Scholar
Louveau, A., Une méthode topologique pour l’étude de la propriété de Ramsey. Israel Journal of Mathematics, vol. 23 (1976), no. 2, 97–116.CrossRefGoogle Scholar
RecŁaw, I., Metric spaces not distinguishing pointwise and quasinormal convergence of real functions. Bulletin of the Polish Academy of Sciences Mathematics, vol. 45 (1997), no. 3, 287–289.Google Scholar
Repický, M., Spaces not distinguishing convergences, Commentationes Mathematicae Universitatis Carolinae, vol. 41 (2000), no. 4, 829–842.Google Scholar
Repický, M., Spaces not distinguishing ideal convergences of real-valued functions, Preprint available from http://im.saske.sk/~repicky/ijwqn-I.pdf.Google Scholar
Repický, M., Spaces not distinguishing ideal convergences of real-valued functions, II, Preprint available from http://im.saske.sk/~repicky/ijwqn-II.pdf.Google Scholar
Sakai, M., The sequence selection properties of ${C}_p(X)$ . Topology and its Applications, vol. 154 (2007), no. 3, 552–560.CrossRefGoogle Scholar
Šottová, V. and Šupina, J., Principle ${S}_1\left(\mathbf{\mathcal{P}},\mathbf{\mathcal{R}}\right)$ : ideals and functions. Topology and its Applications, vol. 258 (2019), 282–304.CrossRefGoogle Scholar
Staniszewski, M., On ideal equal convergence II. Journal of Mathematical Analysis and Applications, vol. 451 (2017), no. 2, 1179–1197.CrossRefGoogle Scholar
Šupina, J., Ideal QN-spaces. Journal of Mathematical Analysis and Applications, vol. 435 (2016), no. 1, 477–491.CrossRefGoogle Scholar
Talagrand, M., Compacts de fonctions mesurables et filtres non mesurables. Studia Mathematica, vol. 67 (1980), no. 1, 13–43.CrossRefGoogle Scholar
Tsaban, B. and Zdomskyy, L., Hereditarily Hurewicz spaces and Arhangel’skiĭ sheaf amalgamations, Journal of the European Mathematical Society (JEMS), vol. 14 (2012), no. 2, 353–372.CrossRefGoogle Scholar
Vojtáš, P., Generalized Galois-Tukey-connections between explicit relations on classical objects of real analysis, Set Theory of the Reals (Ramat Gan, 1991), Israel Mathematics Conference Proceedings, 6, Bar-Ilan University, Ramat Gan, 1993, pp. 619–643.Google Scholar