Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Hurewicz sets of reals without perfect subsets


Authors: Dusan Repovs, Boaz Tsaban and Lyubomyr Zdomskyy
Journal: Proc. Amer. Math. Soc. 136 (2008), 2515-2520
MSC (2000): Primary 37F20; Secondary 26A03, 03E70
DOI: https://doi.org/10.1090/S0002-9939-08-09193-4
Published electronically: March 5, 2008
MathSciNet review: 2390521
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that even for subsets $ X$ of the real line that do not contain perfect sets, the Hurewicz property does not imply the property $ \mathsf{S}_1(\Gamma,\Gamma)$, asserting that for each countable family of open $ \gamma$-covers of $ X$, there is a choice function whose image is a $ \gamma$-cover of $ X$. This settles a problem of Just, Miller, Scheepers, and Szeptycki. Our main result also answers a question of Bartoszyński and the second author, and implies that for $ C_p(X)$, the conjunction of Sakai's strong countable fan tightness and the Reznichenko property does not imply Arhangel$ '$skiı's property $ \alpha_2$.


References [Enhancements On Off] (What's this?)

  • 1. A. V. Arhangel$ '$skiı, The frequency spectrum of a topological space and the classification of spaces, Soviet Mathematical Doklady 13 (1972), 1186-1189. MR 0394575 (52:15376)
  • 2. T. Bartoszyński and B. Tsaban, Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures, Proceedings of the American Mathematical Society 134 (2006), 605-615. MR 2176030 (2006f:54038)
  • 3. A. R. Blass, Combinatorial cardinal characteristics of the continuum, in: Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), Kluwer Academic Publishers, Dordrecht, to appear. http://www.math.lsa.umich.edu/˜ablass/hbk.pdf
  • 4. L. Bukovský and J. Haleš, QN-spaces, wQN-spaces and covering properties, Topology and its Applications 154 (2007), 848-858. MR 2294632
  • 5. L. Bukovský, I. Reclaw, and M. Repický, Spaces not distinguishing pointwise and quasinormal convergence of real functions, Topology and its Applications 41 (1991), 25-41. MR 1129696 (93b:54037)
  • 6. W. Hurewicz, Über Folgen stetiger Funktionen, Fundamenta Mathematicae 9 (1927), 193-204.
  • 7. W. Just, A. W. Miller, M. Scheepers, and P. J. Szeptycki, The combinatorics of open covers, II, Topology and its Applications 73 (1996), 241-266. MR 1419798 (98g:03115a)
  • 8. Lj. D. R. Kočinac and M. Scheepers, Function spaces and a property of Reznichenko, Topology and its Applications 123 (2002), 135-143. MR 1921655 (2003g:54036)
  • 9. Lj. D. R. Kočinac and M. Scheepers, Combinatorics of open covers (VII): Groupability, Fundamenta Mathematicae 179 (2003), 131-155. MR 2029229 (2005c:54020)
  • 10. M. Sakai, Property $ C''$ and function spaces, Proceedings of the American Mathematical Society 104 (1988), 917-919. MR 964873 (89m:54004)
  • 11. M. Sakai, The sequence selection properties of $ C_p(X)$, Topology and its Applications 154 (2007), 552-560. MR 2280899
  • 12. M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology and its Applications 69 (1996), 31-62. MR 1378387 (97h:90123)
  • 13. B. Tsaban and L. Zdomskyy, Scales, fields, and a problem of Hurewicz, Journal of the European Mathematical Society, to appear.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37F20, 26A03, 03E70

Retrieve articles in all journals with MSC (2000): 37F20, 26A03, 03E70


Additional Information

Dusan Repovs
Affiliation: Institute of Mathematics, Physics and Mechanics and Faculty of Education, University of Ljubljana, P.O.B. 2964, Ljubljana, Slovenija 1001
Email: dusan.repovs@guest.arnes.si

Boaz Tsaban
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel; and Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Email: tsaban@math.biu.ac.il

Lyubomyr Zdomskyy
Affiliation: Kurt Gödel Research Center for Mathematical Logic, Währinger Str. 25, A-1090 Vienna, Austria
Email: lzdomsky@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-08-09193-4
Received by editor(s): December 5, 2006
Received by editor(s) in revised form: April 9, 2007
Published electronically: March 5, 2008
Additional Notes: The first and the third authors were supported by the Slovenian Research Agency grants P1-0292-0101-04 and BI-UA/04-06-007.
The second author was partially supported by the Koshland Center for Basic Research.
Communicated by: Julia Knight
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society