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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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