Abstract
We study the class of first-countable Lindelöf scattered spaces, or “FLS” spaces. While every T3 FLS space is homeomorphic to a scattered subspace of Q, the class of T2 FLS spaces turns out to be surprisingly rich. Our investigation of these spaces reveals close ties to Q-sets, Lusin sets, and their relatives, and to the cardinals b and d. Many natural questions about FLS spaces turn out to be independent of ZFC.We prove that there exist uncountable FLS spaces with scattered height ω. On the other hand, an uncountable FLS space with finite scattered height exists if and only if b=ℵ1. We prove some independence results concerning the possible cardinalities of FLS spaces, and concerning what ordinals can be the scattered height of an FLS space. Several open problems are included.
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