Hurewicz sets of reals without perfect subsets

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 $\alpha _2$.