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

Abstract

We prove that every Lindelöf 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].