TY - JOUR
T1 - Hurewicz sets of reals without perfect subsets
AU - Repovš, Dušan
AU - Tsaban, Boaz
AU - Zdomskyy, Lyubomyr
PY - 2008/7
Y1 - 2008/7
N2 - 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 S 1(Γ, Γ), asserting that for each countable family of open γ-covers of X, there is a choice function whose image is a γ-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'skiǐ's property α 2.
AB - 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 S 1(Γ, Γ), asserting that for each countable family of open γ-covers of X, there is a choice function whose image is a γ-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'skiǐ's property α 2.
UR - http://www.scopus.com/inward/record.url?scp=77950790795&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-08-09193-4
DO - 10.1090/S0002-9939-08-09193-4
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AN - SCOPUS:77950790795
SN - 0002-9939
VL - 136
SP - 2515
EP - 2520
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 7
ER -