First-countable Lindel\"of scattered spaces

Abstract

We study the class of first-countable Lindel\"of scattered spaces, or "FLS" spaces. While every $T_3$ FLS space is homeomorphic to a scattered subspace of $\mathbb Q$, the class of $T_2$ 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 $\mathfrak{b}$ and $\mathfrak{d}$. Many natural questions about FLS spaces turn out to be independent of $\mathsf{ZFC}$. We prove that there exist uncountable FLS spaces with scattered height $\omega$. On the other hand, an uncountable FLS space with finite scattered height exists if and only if $\mathfrak{b} = \aleph_1$. 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.