First-countable Lindel\"of scattered spaces

TL;DR

This paper explores the properties and diversity of first-countable Lindel"of scattered spaces, revealing their connections to set-theoretic concepts and demonstrating independence results related to their cardinalities and scattered heights.

Contribution

It provides new insights into the structure of FLS spaces, including existence results, independence from ZFC, and links to set-theoretic cardinals and special sets.

Findings

Existence of uncountable FLS spaces with scattered height ω

Uncountable FLS space with finite scattered height exists iff b =

Many questions about FLS spaces are independent of ZFC

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.