Hurewicz Sets of Reals without Perfect Subsets

We show that even for subsetsX of the real line which do not contain perfect sets, the Hurewicz property does not imply the property S1(Γ,Γ), asserting that for each countable family of open γ-covers of X , there is a choice function whose image is a γcover 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 Cp(X), the conjunction of Sakai’s strong countable fan tightness and the Reznichenko property does not imply Arhangel’skĭı’s property α2.