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Proceedings of the American Mathematical Society
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
Published electronically: March 5, 2008
MathSciNet review: 2390521
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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)$, the conjunction of Sakai's strong countable fan tightness and the Reznichenko property does not imply Arhangel$ '$skiı's property $ \alpha_2$.


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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: http://dx.doi.org/10.1090/S0002-9939-08-09193-4
PII: S 0002-9939(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