Raghavan, Dilip; Steprāns, Juris The almost disjointness invariant for products of ideals. (English) Zbl 07628737 Topology Appl. 323, Article ID 108295, 11 p. (2023). Summary: The almost disjointness numbers associated to the quotients determined by the transfinite products of the ideal of finite sets are investigated. A ZFC lower bound involving the minimum of the classical almost disjointness and splitting numbers is proved for these characteristics. En route, it is shown that the splitting numbers associated to these quotients are all equal to the classical splitting number. Finally, it is proved to be consistent that the almost disjointness numbers associated to these quotients are all equal to the second uncountable cardinal while the bounding number is the first uncountable cardinal. Several open problems are considered. Cited in 1 Document MSC: 03E17 Cardinal characteristics of the continuum 03E35 Consistency and independence results 03E05 Other combinatorial set theory Keywords:cardinal characteristics; definable quotients; almost disjoint families; splitting families; unbounded families PDFBibTeX XMLCite \textit{D. Raghavan} and \textit{J. Steprāns}, Topology Appl. 323, Article ID 108295, 11 p. (2023; Zbl 07628737) Full Text: DOI arXiv References: [1] Blass, A.; Dobrinen, N.; Raghavan, D., The next best thing to a P-point, J. Symb. Log., 80, 3, 866-900 (2015) · Zbl 1367.03078 [2] J. Brendle, Notes on cardinal characteristics of quotients, unpublished notes. Results communicated to first author in January 2021. [3] Brendle, J., Mob families and mad families, Arch. Math. Log., 37, 3, 183-197 (1997) · Zbl 0905.03034 [4] Brendle, J.; Khomskii, Y., Mad families constructed from perfect almost disjoint families, J. Symb. Log., 78, 4, 1164-1180 (2013) · Zbl 1375.03057 [5] Brendle, J.; Raghavan, D., Bounding, splitting, and almost disjointness, Ann. Pure Appl. Log., 165, 2, 631-651 (2014) · Zbl 1405.03081 [6] Brendle, J.; Shelah, S., Ultrafilters on ω—their ideals and their cardinal characteristics, Trans. Am. Math. Soc., 351, 7, 2643-2674 (1999) · Zbl 0927.03073 [7] Grimeisen, G., Ein Approximationssatz für Bairesche Funktionen, Math. Ann., 146, 189-194 (1962) · Zbl 0133.00503 [8] Hernández-Hernández, F.; Hrušák, M., Cardinal invariants of analytic P-ideals, Can. J. Math., 59, 3, 575-595 (2007) · Zbl 1119.03046 [9] Katětov, M., On descriptive classification of functions, (General Topology and Its Relations to Modern Analysis and Algebra, III, Proc. Third Prague Topological Sympos.. General Topology and Its Relations to Modern Analysis and Algebra, III, Proc. Third Prague Topological Sympos., 1971 (1972)), 235-242 · Zbl 0309.54015 [10] Raghavan, D.; Shelah, S., Comparing the closed almost disjointness and dominating numbers, Fundam. Math., 217, 1, 73-81 (2012) · Zbl 1258.03058 [11] Schrittesser, D.; Törnquist, A. [12] Shelah, S., On cardinal invariants of the continuum, (Axiomatic Set Theory. Axiomatic Set Theory, Boulder, CO, 1983. Axiomatic Set Theory. Axiomatic Set Theory, Boulder, CO, 1983, Contemp. Math., vol. 31 (1984), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 183-207 · Zbl 0583.03035 [13] Shelah, S., Two cardinal invariants of the continuum \((\mathfrak{d} < \mathfrak{a})\) and FS linearly ordered iterated forcing, Acta Math., 192, 2, 187-223 (2004) · Zbl 1106.03044 [14] Szymański, A.; Zhou, H. X., The behaviour of \(\omega^{2^\ast}\) under some consequences of Martin’s axiom, (General Topology and Its Relations to Modern Analysis and Algebra, V. General Topology and Its Relations to Modern Analysis and Algebra, V, Prague, 1981. General Topology and Its Relations to Modern Analysis and Algebra, V. General Topology and Its Relations to Modern Analysis and Algebra, V, Prague, 1981, Sigma Ser. Pure Math., vol. 3 (1983), Heldermann: Heldermann Berlin), 577-584 · Zbl 0506.54007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.