On countably selective spaces

Abstract 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 and regular space Y. 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 ( ω + 1 ) -selective space has Arhangel'skii's property α 1 . We obtain an example under p = c of a strongly ( ω + 1 ) -selective space that is not C-selective, and we show that it is consistent with and independent of ZFC that a space is strongly ( ω + 1 ) -selective iff it is ( ω + 1 ) -selective and Frechet. Finally, we answer a question of the third author and Junnila by showing that the ordinal space ω 1 + 1 is not self-selective.