On the Menger covering property and \(D\)-spaces (Q2880668)

A~neighbourhood assignment for a~topological space~\(X\) is a~function \(N:X\to\text{Open}(X)\) such that \(x\in N(x)\) for all \(x\in X\). A~topological space~\(X\) is said to be a~\(D\)-space if for every neighbourhood assignment~\(N\) there is a~closed discrete set \(A\subseteq X\) such that \(X=\bigcup_{x\in A}N(x)\). A~Lindelöf space~\(Y\) is called a~Michael space if \(\omega^\omega\times Y\) is not Lindelöf. The following theorems are the main results of the paper: 1.~It is consistent that every subparacompact space \(X\) of size~\(\omega_1\) is a~\(D\)-space. 2.~If there exists a~Michael space, then all productively Lindelöf spaces have the Menger property and, therefore, are \(D\)-spaces. 3.~Every locally \(D\)-space which admits a~\(\sigma\)-locally finite cover by Lindelöf spaces is a~\(D\)-space.