Totally paracompact spaces and the Menger covering property

TL;DR

This paper investigates the relationship between total paracompactness and the Menger covering property in various classes of topological spaces, providing new equivalences, game-theoretic proofs, and examples related to Lindelöf and Sorgenfrey spaces.

Contribution

It establishes the equivalence of total paracompactness and the Menger property in Lindelöf GO-spaces and introduces game-theoretic methods to analyze these properties.

Findings

Equivalence holds in Lindelöf GO-spaces on subsets of reals.

Regular Menger spaces are totally paracompact via game-theoretic proof.

Existence of uncountable Lindelöf subspaces in Sorgenfrey line when =1.

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 Lindel\"of 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 Lindel\"of 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 $\mathfrak{b}=\omega_1$, then there is an uncountable subspace of the Sorgenfrey line whose all finite powers are Lindel\"of, which is a strengthening of a famous result due to Michael.