Skip to main content
×
×
Home

Combinatorial images of sets of reals and semifilter trichotomy

  • Boaz Tsaban (a1) and Lyubomyr Zdomskyy (a2)
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.

Copyright
Corresponding author
Kurt Gödel Research Center for Mathematical Logic, Währinger Str. 25, A-1090, Vienna, Austria, E-mail: lzdomsky@gmail.com
References
Hide All
[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.
[2]Bartoszyński, T., Shelah, S., and Tsaban, B., Additivity properties of topological diagonalizations, this Journal, vol. 68 (2003), pp. 12541260.
[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.
[4]Blass, A., Groupwise density and related cardinals, Archive for Mathematical Logic, vol. 30 (1990), pp. 111.
[5]Blass, A. and Laflamme, C., Consistency results about filters and the number of inequivalent growth types, this Journal, vol. 54 (1989), pp. 5056.
[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. 213243.
[7]Chaber, J. and Pol, R., A remark on Fremlin-Miller theorem concerning the Menger property and Michael concentrated sets, unpublished note (10 2002).
[8]Galvin, F. and Miller, A., γ-sets and other singular sets of real numbers, Topology and its Applications, vol. 17 (1984). pp. 145155.
[9]Gerlits, J. and Nagy, Zs., Some properties of C(X), I, Topology and its Applications, vol. 14 (1982). pp. 151161.
[10]Just, W., Miller, A., Scheepers, M., and Szeptycki, P., The combinatorics of open covers II, Topology and its Applications, vol. 73 (1996), pp. 241266.
[11]Kočinac, L. and Scheepers, M., Combinatorics of open covers (VII): Groupability, Fundamenta Mathematicae, vol. 179 (2003), pp. 131155.
[12]Laflamme, C., Equivalence of families of functions on the natural numbers, Transactions of the American Mathematical Society, vol. 330 (1992), pp. 307319.
[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. 6071.
[14]Recław, I., Every Luzin set is undetermined in the point-open game, Fundamenta Mathematicae, vol. 144 (1994), pp. 4354.
[15]Sakai, M., Two properties of CP(X) weaker than the Frechet Urysohn property, Topology audits Applications, vol. 153 (2006), pp. 27952804.
[16]Samet, N., Scheepers, M., and Tsaban, B., Partition relations for Hurewicz-type selection hypotheses, Topology and its Applications, to appear.
[17]Scheepers, M., Combinatorics of open covers I: Ramsey theory, Topology and its Applications, vol. 69 (1996), pp. 3162.
[18]Scheepers, M. and Tsaban, B., The combinatorics of Borel covers, Topology and its Applications, vol. 121 (2002), pp. 357382.
[19]Tsaban, B., A diagonalization property between Hurewicz and Menger, Real Analysis Exchange, vol. 27 (2001/2002), pp. 757763.
[20]Tsaban, B., The combinatorics of splittability, Annals of Pure and Applied Logic, vol. 129 (2004), pp. 107130.
[21]Tsaban, B., The Hurewicz covering property and slaloms in the Baire space, Fundamenta Mathematicae, vol. 181 (2004), pp. 273280.
[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. 245282.
[23]Tsaban, B. and Zdomskyy, L., Menger's covering property and groupwise density, this Journal, vol. 71 (2006), pp. 10531056.
[24]Tsaban, B. and Zdomskyy, L., Scales, fields, and a problem of Hurewicz, Journal of the European Mathematical Society, vol. 10 (2008), pp. 837866.
[25]Zdomskyy, L., A semifilter approach to selection principles, Commentationes Mathematicae Universitatis Carolinae, vol. 46 (2005), pp. 525540.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed