| Title:
|
A semifilter approach to selection principles (English) |
| Author:
|
Zdomsky, Lubomyr |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
3 |
| Year:
|
2005 |
| Pages:
|
525-539 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we develop the semifilter approach to the classical Menger and Hurewicz properties and show that the small cardinal $\frak g$ is a lower bound of the additivity number of the $\sigma$-ideal generated by Menger subspaces of the Baire space, and under $\frak u < \frak g$ every subset $X$ of the real line with the property $\operatorname{Split} (\Lambda ,\Lambda )$ is Hurewicz, and thus it is consistent with ZFC that the property $\operatorname{Split} (\Lambda ,\Lambda )$ is preserved by unions of less than $\frak b$ subsets of the real line. (English) |
| Keyword:
|
Menger property |
| Keyword:
|
Hurewicz property |
| Keyword:
|
property $\operatorname{Split}(\Lambda, \Lambda )$ |
| Keyword:
|
semifilter |
| Keyword:
|
multifunction |
| Keyword:
|
small cardinals |
| Keyword:
|
additivity number |
| MSC:
|
03Axx |
| MSC:
|
03E17 |
| MSC:
|
03E35 |
| MSC:
|
54D20 |
| idZBL:
|
Zbl 1121.03060 |
| idMR:
|
MR2174530 |
| . |
| Date available:
|
2009-05-05T16:53:03Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119546 |
| . |
| Related article:
|
http://dml.cz/handle/10338.dmlcz/119614 |
| . |
| Reference:
|
[1] Blass A.: Combinatorial cardinal characteristics of the continuum.in Handbook of Set Theory (M. Foreman et. al., Eds.), to appear. MR 2768685 |
| Reference:
|
[2] Bukovský L., Halež J.: On Hurewicz properties.Topology Appl. 132 (2003), 71-79. MR 1990080 |
| Reference:
|
[3] Blass A., Mildenberger H.: On the cofinality of ultrapowers.J. Symbolic Logic 74 (1999), 727-736. Zbl 0930.03060, MR 1777781 |
| Reference:
|
[4] Bartoszyński T., Shelah S., Tsaban B.: Additivity properties of topological diagonalizations.J. Symbolic Logic 68 (2003), 1254-1260. Zbl 1071.03031, MR 2017353 |
| Reference:
|
[5] Bartoszyński T., Tsaban B.: Hereditary Topological Diagonalizations and the Menger-Hurewicz Conjectures.Proc. Amer. Math. Soc., to appear; http://arxiv.org/abs/math.LO/0208224. MR 2176030 |
| Reference:
|
[6] Banakh T., Zdomsky L.: Coherence of semifilters.submitted; http://www.franko.lviv.ua/faculty/mechmat/ Departments/Topology/booksite.html. |
| Reference:
|
[7] Chaber J., Pol R.: A remark on Fremlin-Miller theorem concerning the Menger property and Michael concentrated sets.preprint. |
| Reference:
|
[8] Gerlits J., Nagy Zs.: Some properties of $C(X)$, I.Topology Appl. 14 2 (1982), 151-163. Zbl 0503.54020, MR 0667661 |
| Reference:
|
[9] Hurewicz W.: Über Folgen stetiger Funktionen.Fund. Math. 9 (1927), 193-204. |
| Reference:
|
[10] Just W., Miller A.W., Scheepers M., Szeptycki P.J.: The combinatorics of open covers II.Topology Appl. 73 (1996), 241-266. Zbl 0870.03021, MR 1419798 |
| Reference:
|
[11] Rogers C.A., Jayne J.E.: $K$-analytic Sets.in Analytic Sets (C.A. Rogers et al., Eds.), Academic Press, 1980, pp.1-179. Zbl 0589.54047 |
| Reference:
|
[12] Kechris A.: Classical Descriptive Set Theory.Graduate Texts in Math. 156, Springer, 1995. Zbl 0819.04002, MR 1321597 |
| Reference:
|
[13] Laflamme C.: Equivalence of families of functions on natural numbers.Trans. Amer. Math. Soc. 330 (1992), 307-319. MR 1028761 |
| Reference:
|
[14] Laflamme C., Leary C.C.: Filter games on $ømega$ and the dual ideal.Fund. Math. 173 (2002), 159-173. Zbl 0998.03038 |
| Reference:
|
[15] Menger K.: Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte.Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie) 133 (1924), 421-444. |
| Reference:
|
[16] Sakai M.: Weak Fréchet-Urysohn property in function spaces.preprint, 2004. |
| Reference:
|
[17] Scheepers M.: Combinatorics of open covers I: Ramsey Theory.Topology Appl. 69 (1996), 31-62. Zbl 0848.54018, MR 1378387 |
| Reference:
|
[18] Talagrand M.: Filtres: Mesurabilité, rapidité, propriété de Baire forte.Studia Math. 74 (1982), 283-291. MR 0683750 |
| Reference:
|
[19] Todorcevic S.: Topics in Topology.Lecture Notes in Math. 1652, Springer, Berlin, 1997. Zbl 0953.54001, MR 1442262 |
| Reference:
|
[20]
: SPM Bulletin 9 (B. Tsaban, Ed.), http://arxiv.org/abs/math.GN/0406411.. |
| Reference:
|
[21] Tsaban B.: The combinatorics of splittability.Ann. Pure Appl. Logic 129 (2004), 107-130; http://arxiv.org/abs/math.GN/0307225. Zbl 1067.03055, MR 2078361 |
| Reference:
|
[22] Tsaban B.: The Hurewicz covering property and slaloms in the Baire space.Fund. Math. 181 (2004), 273-280; http://arxiv.org/abs/math.GN/0301085. Zbl 1056.54028, MR 2099604 |
| Reference:
|
[23] Tsaban B.: Selection principles in mathematics: A milestone of open problems.Note Math. 22 2 (2003/2004), 179-208; http://arxiv.org/abs/math.GN/0312182. MR 2112739 |
| Reference:
|
[24] Tsaban B.: Some new directions in infinite-combinatorial topology.to appear in Topics in Set Theory and its Applications (J. Bagaria and S. Todorcevic, Eds.), Birkhäuser, 2005; http://arxiv.org/abs/math.GN/0409069. Zbl 1113.54002, MR 2267150 |
| Reference:
|
[25] Vaughan J.: Small uncountable cardinals and topology.in Open Problems in Topology (J. van Mill, G.M. Reed, Eds.), Elsevier Sci. Publ., Amsterdam, 1990, pp.197-216. MR 1078636 |
| . |
