Abstract
We study cardinal invariants connected to certain classical orderings on the family of ideals on ω. We give topological and analytic characterizations of these invariants using the idealized version of Fréchet-Urysohn property and, in a special case, using sequential properties of the space of finitely-supported probability measures with the weak* topology. We investigate consistency of some inequalities between these invariants and classical ones, and other related combinatorial questions. At last, we discuss maximality properties of almost disjoint families related to certain ordering on ideals.
Article PDF
Similar content being viewed by others
References
Bell M.G.: On the combinatorial principle p(c). Fund. Math. 114(2), 149–157 (1981) MR 643555 (83e:03077)
Blass A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory, pp. 395–491. Springer, Berlin (2010)
Borodulin-Nadzieja, P., Plebanek, G.: On sequential properties of Banach spaces, spaces of measures and densities. Czechoslovak Math. J. 60(135)(2), 381–399 (2010) MR 2657956
Brendle, J.: Cardinal invariants of analytic quotient. Slides for ESI workshop on large cardinals and descriptive set theory, Vienna, June 14–27 (2009)
Brendle J., Shelah S.: Ultrafilters on ω—their ideals and their cardinal characteristics. Transactions of the American Mathematical Society 351, 2643–2674 (1999) math.LO/9710217
Engelking, R., Karłowicz, M.: Some theorems of set theory and their topological consequences. Fund. Math. 57, 275–285 (1965) MR 0196693 (33 #4880)
Fremlin, D.H.: Measure theory: Topological measure spaces v. 4. Torres Fremlin, (2003)
Hernández-Hernández F., Hrušák M.: Cardinal invariants of analytic P-ideals. Canad. J. Math. 59(3), 575–595 (2007) MR 2319159 (2008c:03051)
Hrušák, M.: Combinatorics of filters and ideals. In: Set Theory and its Applications. Contemporary Mathematics, vol. 533, pp. 29–69. American Mathematical Society, Providence, RI (2011)
Hrušák, M., Minami, H.: Mathias-Prikry and Laver-Prikry type forcing. Preprint (2010)
Mazur, K.: F σ -ideals and ω 1 ω *1 -gaps in the Boolean algebras P(ω)/I. Fund. Math. 138(2), 103–111 (1991) MR 1124539 (92g:06019)
Meza-Alcántara, D.: Ideals and filters on countable sets. Ph.D. thesis, Universidad Nacional Autónoma México, (2009)
Niederreiter H.: On the existence of uniformly distributed sequences in compact spaces. Compositio Math. 25, 93–99 (1972)
Solecki S.: Analytic ideals and their applications. Ann. Pure Appl. Logic 99(1–3), 51–72 (1999) MR 1708146 (2000g:03112)
Solomon, R.C.: Families of sets and functions, Czechoslovak Math. J. 27(102)(4), 556–559 (1977) MR 0457218 (56 #15429)
Acknowledgements
A part of the paper was born in a fertile atmosphere of the Winter school on Abstract Analysis (Section Topology) 2010 in Hejnice, Czech Republic. The authors want to thank the occupants of the smokers room in Hejnice, particularly Prof. Petr Simon and Prof. Bohuslav Balcar, for interesting discussions about the subject of this paper. We would also like to thank Prof. Andreas Blass for his valuable remarks on the Katětov-order.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was partially supported by the grant N N201 418939 from the Polish Ministry of Science and Higher Education and the grant 2351/W/IM/10 from University of Wrocław. The second author was supported by Hungarian National Foundation for Scientific Research grant nos. 68262 and 77476.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Borodulin-Nadzieja, P., Farkas, B. Cardinal coefficients associated to certain orders on ideals. Arch. Math. Logic 51, 187–202 (2012). https://doi.org/10.1007/s00153-011-0260-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-011-0260-9
Keywords
- Katětov-order
- Analytic ideals
- Cardinal invariants of the continuum
- Almost disjoint families
- Frechet-Urysohn property
- Weak* topology