Every Lindelöf scattered subspace of a $Σ$-product of real lines is $σ$-compact

TL;DR

This paper addresses whether Lindelöf scattered subspaces embedded in $\\Sigma$-products of first-countable spaces must be $\\sigma$-compact. It develops an inductive argument on the Cantor-Bendixson rank $ht(X)$, aided by a new Alster-type lemma, to obtain open refinements whose closures intersecting $X$ are $\\sigma$-compact. The main theorem asserts that if $X$ is a Lindelöf scattered subspace of a $\\Sigma$-product $\\Sigma(\\prod_{\\gamma\in \Gamma} Y_\\gamma, a)$ with each $Y_\\gamma$ first-countable, then $X$ is $\\sigma$-compact; this yields that Lindelöf scattered subspaces of Corson compacta and, in particular, of Eberlein compacta are $\\sigma$-compact. The results resolve questions posed by Tkachenko and colleagues and have implications for the structure of $\\Delta$-spaces arising from Such $\\Sigma$-products.

Abstract

We prove that every Lindelöf scattered subspace of a $Σ$-product of first-countable spaces is $σ$-compact. In particular, we obtain the result stated in the title. This answers some questions of Tkachuk from [Houston J. Math. 48 (2022), no. 1, 171--181].

Theorems & Definitions (15)

• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Claim 1
• proof