Hostname: page-component-76c49bb84f-ckhzl Total loading time: 0 Render date: 2025-07-09T09:44:52.468Z Has data issue: false hasContentIssue false
  • English
  • Français

Combinatorial images of sets of reals and semifilter trichotomy

Published online by Cambridge University Press:  12 March 2014

Boaz Tsaban  and
Lyubomyr Zdomskyy
Boaz Tsaban
Affiliation:
Bar-Ilan University, Department of Mathematics, Ramat-Gan 52900, Israeland, Weizmann Institute of Science, Department of Mathematics, Rehovot 76100, Israel, E-mail: tsaban@math.biu.ac.il, URL: http://www.cs.biu.ac.il/~tsaban
Lyubomyr Zdomskyy*
Affiliation:
Ivan Franko Lviv National University, Department of Mechanics and Mathematics, Universytetska 1, LVIV 79000, Ukraine, and, Weizmann Institute of Science, Department of Mathematics, Rehovot 76100, Israel
*
Kurt Gödel Research Center for Mathematical Logic, Währinger Str. 25, A-1090, Vienna, Austria, E-mail: lzdomsky@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Using a dictionary translating a variety of classical and modern covering properties into combinatorial properties of continuous images, we get a simple way to understand the interrelations between these properties in ZFC and in the realm of the trichotomy axiom for upward closed families of sets of natural numbers. While it is now known that the answer to the Hurewicz 1927 problem is positive, it is shown here that semifilter trichotomy implies a negative answer to a slightly stronger form of this problem.

Keywords

scheepers propertysemifilter trichotomy

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1278 - 1288
Copyright
Copyright © Association for Symbolic Logic 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Banakh, T. and Zdomskyy, L.. Coherence of semifilters. Preliminary version of the book available from: www.franko.lviv.ua/faculty/mechmat/Departments/Topology/booksite.html.Google Scholar
[2]Bartoszyński, T., Shelah, S., and Tsaban, B., Additivity properties of topological diagonalizations, this Journal, vol. 68 (2003), pp. 1254–1260.Google Scholar
[3]Blass, A., Combinatorial cardinal characteristics of the continuum, Handbook of Set Theory (Foreman, M., Kanamori, A., and Magidor, M., editors), Kluwer Academic Publishers, to appear.Google Scholar
[4]Blass, A., Groupwise density and related cardinals, Archive for Mathematical Logic, vol. 30 (1990), pp. 1–11.CrossRefGoogle Scholar
[5]Blass, A. and Laflamme, C., Consistency results about filters and the number of inequivalent growth types, this Journal, vol. 54 (1989), pp. 50–56.Google Scholar
[6]Blass, A. and Shelah, S., There may be simple , - and -points, and the Rudin-Keisler ordering may be downward directed, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 213–243.CrossRefGoogle Scholar
[7]Chaber, J. and Pol, R., A remark on Fremlin-Miller theorem concerning the Menger property and Michael concentrated sets, unpublished note (10 2002).Google Scholar
[8]Galvin, F. and Miller, A., γ-sets and other singular sets of real numbers, Topology and its Applications, vol. 17 (1984). pp. 145–155.CrossRefGoogle Scholar
[9]Gerlits, J. and Nagy, Zs., Some properties of C(X), I, Topology and its Applications, vol. 14 (1982). pp. 151–161.CrossRefGoogle Scholar
[10]Just, W., Miller, A., Scheepers, M., and Szeptycki, P., The combinatorics of open covers II, Topology and its Applications, vol. 73 (1996), pp. 241–266.CrossRefGoogle Scholar
[11]Kočinac, L. and Scheepers, M., Combinatorics of open covers (VII): Groupability, Fundamenta Mathematicae, vol. 179 (2003), pp. 131–155.CrossRefGoogle Scholar
[12]Laflamme, C., Equivalence of families of functions on the natural numbers, Transactions of the American Mathematical Society, vol. 330 (1992), pp. 307–319.CrossRefGoogle Scholar
[13]Mildenberger, M., Shelah, S., and Tsaban, B., Covering the Baire space by families which are not finitely dominating, Annals of Pure and Applied Logic, vol. 140 (2006), pp. 60–71.CrossRefGoogle Scholar
[14]Recław, I., Every Luzin set is undetermined in the point-open game, Fundamenta Mathematicae, vol. 144 (1994), pp. 43–54.CrossRefGoogle Scholar
[15]Sakai, M., Two properties of CP(X) weaker than the Frechet Urysohn property, Topology audits Applications, vol. 153 (2006), pp. 2795–2804.CrossRefGoogle Scholar
[16]Samet, N., Scheepers, M., and Tsaban, B., Partition relations for Hurewicz-type selection hypotheses, Topology and its Applications, to appear.Google Scholar
[17]Scheepers, M., Combinatorics of open covers I: Ramsey theory, Topology and its Applications, vol. 69 (1996), pp. 31–62.CrossRefGoogle Scholar
[18]Scheepers, M. and Tsaban, B., The combinatorics of Borel covers, Topology and its Applications, vol. 121 (2002), pp. 357–382.CrossRefGoogle Scholar
[19]Tsaban, B., A diagonalization property between Hurewicz and Menger, Real Analysis Exchange, vol. 27 (2001/2002), pp. 757–763.CrossRefGoogle Scholar
[20]Tsaban, B., The combinatorics of splittability, Annals of Pure and Applied Logic, vol. 129 (2004), pp. 107–130.CrossRefGoogle Scholar
[21]Tsaban, B., The Hurewicz covering property and slaloms in the Baire space, Fundamenta Mathematicae, vol. 181 (2004), pp. 273–280.CrossRefGoogle Scholar
[22]Tsaban, B., Additivity numbers of covering properties, Selection Principles and Covering Properties in Topology (Kočinac, L., editor), Quaderni di Matematica, vol. 18, Seconda Universita di Napoli, Caserta, 2007, pp. 245–282.Google Scholar
[23]Tsaban, B. and Zdomskyy, L., Menger's covering property and groupwise density, this Journal, vol. 71 (2006), pp. 1053–1056.Google Scholar
[24]Tsaban, B. and Zdomskyy, L., Scales, fields, and a problem of Hurewicz, Journal of the European Mathematical Society, vol. 10 (2008), pp. 837–866.CrossRefGoogle Scholar
[25]Zdomskyy, L., A semifilter approach to selection principles, Commentationes Mathematicae Universitatis Carolinae, vol. 46 (2005), pp. 525–540.Google Scholar