Continuous Selections of Lower semicontinuous Set-valued Mappings

A space $X$ is strongly $Y$-selective (resp., $Y$-selective) if every lower semicontinuous mapping from $Y$ to the nonempty subsets (resp., nonempty closed subsets) of $X$ has a continuous selection. We also call $X$ (strongly) $C$-selective if it is (strongly) $Y$-selective for any countable space $Y$, and (strongly) $L$-selective if it is (strongly) ($\omega+1$)-selective. E. Michael showed that every first countable space is strongly $C$-selective. We extend this by showing that every $W$-space in the sense of the second author is strongly $C$-selective. We also show that every GO-space is $C$-selective, and that every $L$-selective space has Arhangel'skii's property $\alpha_1$. We obtain an example under $\mathfrak p=\mathfrak c$ of a strongly $L$-selective space that is not $C$-selective, and we show that it is consistent with and independent of ZFC that a space is strongly $L$-selective iff it is $L$-selective and Fr\'echet. Finally, we answer a question of the third author and Junnila by showing that the ordinal space $\omega_1 +1$ is not self-selective.