Every Lindel\"of scattered subspace of a $\Sigma$-product of real lines is $\sigma$-compact

TL;DR

The paper proves that Lindel"of scattered subspaces within certain product spaces are always $\sigma$-compact, answering previously open questions in topology.

Contribution

It establishes that all Lindel"of scattered subspaces of $\Sigma$-products of first-countable spaces are $\sigma$-compact, providing a significant advancement in topological space classification.

Findings

Lindel"of scattered subspaces are $\sigma$-compact

Answers to questions posed by Tkachuk

Applicable to $\Sigma$-products of first-countable spaces

Abstract

We prove that every Lindel\"of scattered subspace of a $\Sigma$-product of first-countable spaces is $\sigma$-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].