Critical cardinalities and additivity properties of combinatorial notions of smallness

• Autorzy: Shelah S. , Tsaban B.
• Języki publikacji: EN

Abstrakty

EN

Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (r-covers). We deal with two types of combinatorial questions which arise from this study. 1. Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. 2. We study the additivity numbers of the combinatorial notions corresponding to the topological diagonalization notions. This gives new insights on the structure of the eventual dominance ordering on the Baire space, the almost inclusion ordering on the Rothberger space, and the interactions between them.

Słowa kluczowe

EN

r-cover; tower; splitting number; additivity number

PL

wieża; rozłożona liczba; liczba addytywności

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2003
• Tom: Vol. 9, nr 2
• Strony: 149--162
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Shelah S.

• Institute of Mathematics Hebrew University of Jerusalem Givat Ram, 91904 Jerusalem Israel and Mathematics Department Rutgers University New Brunswick, NJ 08903 USA

autor

Tsaban B.

• Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel

Bibliografia

• [1] Bartoszyński, T., Judah, H., Set Theory. On the Structure of the Real Line, A К Peters, Wellesley, MA, 1995.
• [2] Bartoszyński, T., Shelah, S., Tsaban, B., Additivity properties of topological diagonalizations, J. Symbolic Logic, (to appear). (Full version: http://arxiv.org/abs/math.L0/0112262)
• [3] Blass, A. R., Combinatorial cardinal characteristics of the continuum, to appear in “Handbook of Set Theory”, (eds. M. Foreman, et. al.).
• [4] Blass, A. R., Mildenberger, H., On the cofinality of ultrapowers, J. Symbolic Logic 64 (1999), 727-736.
• [5] Blass, A. R., Shelah, S., There may be simple Pn1 - and Рn2 - points, and the Rudin Keisler ordering may be downward directed, Ann. Pure Appl. Logic 33 (1987), 213- 243.
• [6] Canjar, R. M., Countable ultraproducts without CH, Ann. Pure Appl. Logic 37 (1988), 1-79.
• [7] Canjar, R. M., Cofinalities of countable ultraproducts: the existence theorem, Notre Dame J. Formal Logic 30 (1989), 539-542.
• [8] Kamburelis, A., Węglorz, В., Splittings, Arch. Math. Logic 35 (1996), 263-277.
• [9] Kunen, K., Inaccessibility Properties of Cardinals, Doctoral Dissertation, Stanford, 1968.
• [10] Mildenberger, H., Groupwise dense families, Arch. Math. Logic 40 (2001), 93-112.
• [11] Roitman, J., Non-isomorphic H-fields from non-isomorphic ultrapowers, Math. Z. 181 (1982), 93-96.
• [12] Rothberger, F., On some problems of Hausdorff and of Sierpiński, Fund. Math. 35 (1948), 29-46.
• [13] Scheepers, M., Tsaban, B., The combinatorics of Borel covers, Topology Appl. 121 (2002), 357-382.
• [14] Shelah, S., Tsaban, B., Critical cardinalities and additivity properties of combinatorial notions of smallness (online version), http://arxiv.org/abs/math.L0/0304019.
• [15] Tsaban, B., A topological interpretation of t1 Real Anal. Exchange 25 (1999/2000), 391-404.
• [16] Tsaban, B., A diagonalization property between Hurewicz and Menger, Real Anal. Exchange 27 (2001/2002), 1-7.
• [17] Tsaban, B., Selection principles and the minimal tower problem, Note di Mat., (to appear).