On the Menger covering property and $D$-spaces

TL;DR

This paper explores conditions under which certain topological spaces possess the $D$-space property, linking it to the Menger property, productively Lindel"of spaces, and the existence of Michael spaces, with results depending on set-theoretic assumptions.

Contribution

It establishes new conditions and consistency results connecting $D$-spaces, the Menger property, and productively Lindel"of spaces, advancing understanding of their interrelations.

Findings

Every subparacompact space of size $oldsymbol{\omega_1}$ can be a $D$-space under certain set-theoretic assumptions.

If a Michael space exists, then all productively Lindel"of spaces are $D$-spaces.

Locally $D$-spaces with $oldsymbol{\sigma}$-locally finite Lindel"of covers are $D$-spaces.

Abstract

The main results of this note are: It is consistent that every subparacompact space $X$ of size $\omega_1$ is a $D$-space; If there exists a Michael space, then all productively Lindel\"of spaces have the Menger property, and, therefore, are $D$-spaces; and Every locally $D$-space which admits a $\sigma$-locally finite cover by Lindel\"of spaces is a $D$-space.