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Infinite dimensional sequential compactness: Sequential compactness based on barriers

Part of: Fairly general properties Maps and general types of spaces defined by maps Generalities Set theory

Published online by Cambridge University Press:  09 December 2024

C. Corral Open the ORCID record for C. Corral [Opens in a new window] ,
O. Guzmán ,
C. López-Callejas Open the ORCID record for C. López-Callejas [Opens in a new window] ,
P. Memarpanahi ,
P. Szeptycki Open the ORCID record for P. Szeptycki [Opens in a new window]  and
S. Todorčević
C. Corral*
Affiliation:
York University, Toronto, Canada
O. Guzmán
Affiliation:
Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Morelia, México e-mail: oguzman@matmor.unam.mx
C. López-Callejas
Affiliation:
Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Morelia, México e-mail: carloscallejas@matmor.unam.mx
P. Memarpanahi
Affiliation:
York University, Toronto, Canada e-mail: pourya7@yorku.ca
P. Szeptycki
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Canada e-mail: szeptyck@yorku.ca
S. Todorčević
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada e-mail: stevo@math.toronto.edu Institut de Mathématiques de Jussieu, CNRS, Paris, France e-mail: stevo.todorcevic@imj-prg.fr Matematički Institut, SANU, Belgrade, Serbia e-mail: stevo.todorcevic@sanu.ac.rs
*
e-mail: cicorral@yorku.ca
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Abstract

We introduce a generalization of sequential compactness using barriers on $\omega $ extending naturally the notion introduced in [W. Kubiś and P. Szeptycki, On a topological Ramsey theorem, Canad. Math. Bull., 66 (2023), 156–165]. We improve results from [C. Corral and O. Guzmán and C. López-Callejas, High dimensional sequential compactness, Fund. Math.] by building spaces that are ${\mathcal {B}}$-sequentially compact but not ${\mathcal {C}}$-sequentially compact when the barriers ${\mathcal {B}}$ and ${\mathcal {C}}$ satisfy certain rank assumption which turns out to be equivalent to a Katětov-order assumption. Such examples are constructed under the assumption ${\mathfrak {b}} ={\mathfrak {c}}$. We also exhibit some classes of spaces that are ${\mathcal {B}}$-sequentially compact for every barrier ${\mathcal {B}}$, including some classical classes of compact spaces from functional analysis, and as a byproduct, we obtain some results on angelic spaces. Finally, we introduce and compute some cardinal invariants naturally associated to barriers.

Keywords

ℬ-sequentially compact spacebarriersequentially compactRamsey convergencealmost disjoint familybounding numberNash-Williamsbisequential

MSC classification

Primary: 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.) 03E17: Cardinal characteristics of the continuum 54D30: Compactness
Secondary: 03E02: Partition relations 54D80: Special constructions of spaces (spaces of ultrafilters, etc.) 54C35: Function spaces
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 28
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The first author acknowledges support from York University and the Fields Institute. The research of the second author was supported by PAPIIT grant IA 104124 and CONAHCYT grant CBF2023-2024-903. The research of the third author was supported by PAPIIT grant IN101323 and CONACyT grant A1-S-16164. The fifth author acknowledges support from NSERC. The research of the sixth author was partially supported by grants from NSERC(455916), CNRS(UMR7586), SFRS(7750027-SMART), and EXPRO 20-31529X (Czech Science Foundation)

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