This paper addresses several questions of Feng, Gruenhage, and Shen which arose from Michael's theory of continuous selections from countable spaces. We construct an example of a space which is $L$-selective but not $\mathbb{ Q}$-selective from $\mathfrak{ d}=\omega_1$, and an $L$-selective space which is not selective for a $P$-point ultrafilter from the assumption of $\mathsf{ CH}$. We also produce $\mathsf{ ZFC}$ examples of Fr\'echet spaces where countable subsets are first countable which are not $L$-selective.
Categories: math.GN math.LO