Totally paracompact spaces and the Menger covering property
Davide Giacopello, Maddalena Bonanzinga, Piotr Szewczak
TL;DR
This paper studies the relationship between total paracompactness and the Menger covering property across several natural classes, including Lindelöf spaces with dense $σ$-compact subsets and GO-spaces on subsets of the real line. It provides a direct game-theoretic proof that every regular $\mathsf{Menger}$ space is totally paracompact and demonstrates that in Lindelöf real GO-spaces, $\mathsf{Menger}$ is equivalent to total paracompactness (and to total metacompactness). It analyzes Aurichi's partial open neighborhood assignment game, showing its equivalence to the Menger game in first-countable and related contexts and proving that totally paracompact spaces are $D$-spaces. The paper also discusses a set-theoretic construction yielding uncountable Sorgenfrey subspaces with Lindelöf finite powers under $\mathfrak{b}=\omega_1$, and closes with open problems about extending these equivalences and the Curtis-type implications.
Abstract
A topological space is totally paracompact if any base of this space contains a locally finite subcover. We focus on a problem of Curtis whether in the class of regular Lindelöf spaces total paracompactness is equivalent to the Menger covering property. To this end we consider topological spaces with certain dense subsets. It follows from our results that the above equivalence holds in the class of Lindelöf GO-spaces defined on subsets of reals. We also provide a game-theoretical proof that any regular Menger space is totally paracompact and show that in the class of first-countable spaces the Menger game and a partial open neighborhood assignment game of Aurichi are equivalent. We also show that if $\mathfrak{b}=ω_1$, then there is an uncountable subspace of the Sorgenfrey line whose all finite powers are Lindelöf, which is a strengthening of a famous result due to Michael.