Hurewicz sets of reals without perfect subsets
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- by Dušan Repovš, Boaz Tsaban and Lyubomyr Zdomskyy
- Proc. Amer. Math. Soc. 136 (2008), 2515-2520
- DOI: https://doi.org/10.1090/S0002-9939-08-09193-4
- Published electronically: March 5, 2008
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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$’$skiĭ’s property $\alpha _2$.References
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Bibliographic Information
- Dušan Repovš
- Affiliation: Institute of Mathematics, Physics and Mechanics and Faculty of Education, University of Ljubljana, P.O.B. 2964, Ljubljana, Slovenija 1001
- MR Author ID: 147135
- ORCID: 0000-0002-6643-1271
- 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
- MR Author ID: 632515
- Email: tsaban@math.biu.ac.il
- Lyubomyr Zdomskyy
- Affiliation: Kurt Gödel Research Center for Mathematical Logic, Währinger Str. 25, A-1090 Vienna, Austria
- MR Author ID: 742789
- Email: lzdomsky@gmail.com
- 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
- © Copyright 2008 American Mathematical Society
- 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
- MathSciNet review: 2390521