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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)

References

Spiros, S. A. and Todorčević, S., Ramsey methods in analysis, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2005, viii–257. https://doi.org/10.1007/3-7643-7360-1 Google Scholar
Blass, A., Combinatorial cardinal characteristics of the continuum . In Handbook of set theory. Vols. 1, 2, 3, 2010, 395–489. https://doi.org/10.1007/978-1-4020-5764-9_7 CrossRefGoogle Scholar
Bojańczyk, M., Kopczyński, E. and Toruńczyk, S., Ramsey’s theorem for colors from a metric space . Semigroup Forum 85(2012), 182–184. https://doi.org/10.1007/s00233-012-9404-4 CrossRefGoogle Scholar
Booth, D., Ultrafilters on a countable set . Ann. Math. Logic 2(1970/71), 1–24. https://doi.org/10.1016/0003-4843(70)90005-7 CrossRefGoogle Scholar
Bourgain, J., Fremlin, D. H. and Talagrand, M., Pointwise compact sets of Baire-measurable functions . Amer. J. Math. 100(1978), 845–886. https://doi.org/10.2307/2373913 CrossRefGoogle Scholar
Cieśla, T., A filter on a collection of finite sets and Eberlein compacta , Topology Appl. 229(2017), 106–111. https://doi.org/10.1016/j.topol.2017.07.011 CrossRefGoogle Scholar
Corral, C., Guzmán, O. and López-Callejas, C., High dimensional sequential compactness . Fund. Math. 264(2024), 21–54. https://doi.org/10.4064/fm230606-11-8 CrossRefGoogle Scholar
Corral, C. and Hrušák, M., Fréchet-like properties and almost disjoint families . Topology Appl. 277(2020). https://doi.org/10.1016/j.topol.2020.107216 CrossRefGoogle Scholar
Debs, G., Effective properties in compact sets of Borel functions . Mathematika 34(1987), 64–68. https://doi.org/10.1112/s0025579300013280 CrossRefGoogle Scholar
Dobrinen, N., Continuous and other finitely generated canonical cofinal maps on ultrafilters . Fund. Math. 249(2020), 111–147. https://doi.org/10.4064/fm691-6-2019 CrossRefGoogle Scholar
van Douwen, E. K., The integers and topology . In Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, 111–167. https://doi.org/10.1016/b978-0-444-86580-9.50006-9 CrossRefGoogle Scholar
Engelking, R., General topology, Monografie Matematyczne, 1977.Google Scholar
Filipów, R., Kowitz, K. and Kwela, A., A unified approach to Hindman, Ramsey and van der Waerden spaces . J. Symb. Log. (2024). https://doi.org/10.1017/jsl.2024.8 CrossRefGoogle Scholar
Hernández-Hernández, F. and Hrušák, M., Topology of Mrówka-Isbell spaces . In Pseudocompact topological spaces, 2018, 253–289. https://doi.org/10.1007/978-3-319-91680-4_8 CrossRefGoogle Scholar
Knaust, H., Array convergence of functions of the first Baire class . Proc. Amer. Math. Soc. 112(1991), 529–532. https://doi.org/10.1090/s0002-9939-1991-1057955-9 CrossRefGoogle Scholar
Knaust, H., Angelic spaces with the Ramsey property . In Interaction between functional analysis, harmonic analysis, and probability, 1994, 267–273.Google Scholar
Hrušák, M., Katětov order on Borel ideals . Arch. Math. Logic. 56(2017), 831–847. https://doi.org/10.1007/s00153-017-0543-x CrossRefGoogle Scholar
Katětov, M., Products of filters , Comment. Math. Univ. Carolinae 9(1968), 173–189.Google Scholar
Kubiś, W. and Szeptycki, P., On a topological Ramsey theorem , Canad. Math. Bull. 66(2023), 156–165. https://doi.org/10.4153/s0008439522000170 CrossRefGoogle Scholar
Kunen, K., Set theory-An introduction to independence proofs, North-Holland Publishing Co., Amsterdam-New York, 1980. https://doi.org/10.1016/s0049-237x(08)x7037-5 Google Scholar
Michael, E. A., A quintuple quotient quest , General Topology and Appl. 2(1972), 91–138. https://doi.org/10.1016/0016-660x(72)90040-2 CrossRefGoogle Scholar
Nash-Williams, C. St. J. A., On well-quasi-ordering transfinite sequences . Proc. Cambridge Philos. Soc. 61(1965), 33–39. https://doi.org/10.1017/s0305004100038603 CrossRefGoogle Scholar
Negrepontis, S., Banach spaces and topology. In Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, 1045–1142. https://doi.org/10.1016/B978-0-444-86580-9.50026-4 CrossRefGoogle Scholar
Pol, R., Note on pointwise convergence of sequences of analytic sets . Mathematika 36(1989), no. 2, 290–300. https://doi.org/10.1112/s0025579300013140 CrossRefGoogle Scholar
Todorčević, S., Compact subsets of the first Baire class . J. Amer. Math. Soc. 12(1999), 1179–1212. https://doi.org/10.1090/s0894-0347-99-00312-4 CrossRefGoogle Scholar
Todorčević, S., Introduction to Ramsey spaces . Ann. of Math. Stud. 174(2010), viii–287. https://doi.org/10.1515/9781400835409Google Scholar