Consistency and independence phenomena involving cellular-Lindelöf spaces

Hernández-Gutiérrez, Rodrigo; Spadaro, Santi

doi:10.1090/proc/17224

Consistency and independence phenomena involving cellular-Lindelöf spaces
HTML articles powered by AMS MathViewer

by Rodrigo Hernández-Gutiérrez and Santi Spadaro;

Proc. Amer. Math. Soc. 153 (2025), 2701-2711

DOI: https://doi.org/10.1090/proc/17224

Published electronically: April 8, 2025

HTML | PDF | Request permission

Abstract:

The cellular-Lindelöf property is a common generalization of the Lindelöf property and the countable chain condition that was introduced by Bella and Spadaro [Monatsh. Math. 186 (2018), pp. 345–353]. We solve two questions of Alas, Gutiérrez-Domínguez and Wilson [Acta Math. Hungar. 167 (2022), pp 548–560] by constructing consistent examples of a normal almost cellular-Lindelöf space which is neither cellular-Lindelöf nor weakly Lindelöf and a Tychonoff cellular-Lindelöf space of Lindelöf degree $\omega _1$ and uncountable weak Lindelöf degree for closed sets. We also construct a ZFC example of a space for which both the almost cellular-Lindelöf property and normality are undetermined in ZFC.

References

O. T. Alas, More topological cardinal inequalities, Colloq. Math. 65 (1993), no. 2, 165–168. MR 1240163, DOI 10.4064/cm-65-2-165-168
O. T. Alas, L. E. Gutiérrez-Domínguez, and R. G. Wilson, Compact productivity of Lindelöf-type properties, Acta Math. Hungar. 167 (2022), no. 2, 548–560. MR 4487625, DOI 10.1007/s10474-022-01254-x
Ofelia T. Alas, Lucia R. Junqueira, Marcelo D. Passos, and Richard G. Wilson, On cellular-compactness and related properties, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (2020), no. 2, Paper No. 101, 13. MR 4076721, DOI 10.1007/s13398-020-00833-3
A. V. Arhangel’skii, A theorem on cardinality, Russ. Math. Surv. 34 (1979), 153–154.
A. V. Arhangel’skii, Topological function spaces, Mathematics and its Applications, Soviet series 78, Kluwer Academic Publishers, Dordrecht, 1992, pp. ix, 205.
Murray Bell, John Ginsburg, and Grant Woods, Cardinal inequalities for topological spaces involving the weak Lindelof number, Pacific J. Math. 79 (1978), no. 1, 37–45. MR 526665
Angelo Bella, On cellular-compact and related spaces, Topology Appl. 281 (2020), 107203, 6. MR 4174603, DOI 10.1016/j.topol.2020.107203
Angelo Bella and Santi Spadaro, On the cardinality of almost discretely Lindelöf spaces, Monatsh. Math. 186 (2018), no. 2, 345–353. MR 3808659, DOI 10.1007/s00605-017-1112-4
Angelo Bella and Santi Spadaro, Cardinal invariants of cellular-Lindelöf spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 2805–2811. MR 3956284, DOI 10.1007/s13398-019-00660-1
Angelo Bella and Santi Spadaro, A common extension of Arhangel’skĭ’s theorem and the Hajnal-Juhász inequality, Canad. Math. Bull. 63 (2020), no. 1, 197–203. MR 4059816, DOI 10.4153/s0008439519000420
Angelo Bella and Santi Spadaro, Strongly discrete subsets with Lindelöf closures, Topology Proc. 59 (2022), 89–98. MR 4216107
Alan Dow and R. M. Stephenson Jr., Productivity of cellular-Lindelöf spaces, Topology Appl. 290 (2021), Paper No. 107606, 15. MR 4199849, DOI 10.1016/j.topol.2021.107606
Vera Fischer, Diana Carolina Montoya, Jonathan Schilhan, and Dániel T. Soukup, Towers and gaps at uncountable cardinals, Fund. Math. 257 (2022), no. 2, 141–166. MR 4391475, DOI 10.4064/fm109-9-2021
J. R. Isbell, Uniform spaces, Mathematical Surveys, No. 12, American Mathematical Society, Providence, RI, 1964. MR 170323
István Juhász, Cardinal functions in topology—ten years later, 2nd ed., Mathematical Centre Tracts, vol. 123, Mathematisch Centrum, Amsterdam, 1980. MR 576927
I. Juhász, L. Soukup, and Z. Szentmiklóssy, On cellular-compact spaces, Acta Math. Hungar. 162 (2020), no. 2, 549–556. MR 4173314, DOI 10.1007/s10474-020-01035-4
M. Malliaris and S. Shelah, Cofinality spectrum theorems in model theory, set theory, and general topology, J. Amer. Math. Soc. 29 (2016), no. 1, 237–297. MR 3402699, DOI 10.1090/jams830
Jonathan Schilhan, Generalised pseudointersections, MLQ Math. Log. Q. 65 (2019), no. 4, 479–489. MR 4057946, DOI 10.1002/malq.201800084
Sumit Singh, On star-cellular-Lindelöf spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (2021), no. 4, Paper No. 190, 12. MR 4314231, DOI 10.1007/s13398-021-01144-x
V. V. Tkachuk and R. G. Wilson, Cellular-compact spaces and their applications, Acta Math. Hungar. 159 (2019), no. 2, 674–688. MR 4022157, DOI 10.1007/s10474-019-00968-9
Boaz Tsaban, Menger’s and Hurewicz’s problems: solutions from “the book” and refinements, Set theory and its applications, Contemp. Math., vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 211–226. MR 2777750, DOI 10.1090/conm/533/10509
V. V. Uspenskiĭ, Continuous images of Lindelöf topological groups, Dokl. Akad. Nauk SSSR 285 (1985), no. 4, 824–827 (Russian). MR 821360
Wei-Feng Xuan and Yan-Kui Song, A study of cellular-Lindelöf spaces, Topology Appl. 251 (2019), 1–9. MR 3872897, DOI 10.1016/j.topol.2018.10.008
Wei-Feng Xuan and Yan-Kui Song, More on cellular-Lindelöf spaces, Topology Appl. 266 (2019), 106861, 12. MR 3997193, DOI 10.1016/j.topol.2019.106861

AMS Website Down

Proceedings of the American Mathematical Society

Consistency and independence phenomena involving cellular-Lindelöf spaces
HTML articles powered by AMS MathViewer

Abstract: