The \(D\)-property of some Lindelöf spaces and related conclusions
All spaces are \(T_1\)-spaces. A space \(X\) is called a \(D\)-space if for any neighborhood assignment \(\Phi\) for \(X\) there exists a closed discrete subset \(D\) of \(X\) such that \(X=\bigcup \{\phi(d):d\in D\}.\) A generalization of this concept is the notion of an \(\alpha D\)-space due to Arhangel'skiĭ and Buzyakova. One of the central questions on \(D\)-spaces posed by van Douwen is whether every Lindelöf space is a \(D\)-space. In the present article the author shows that the Lindelöf space \({\mathcal C}_p(\tau_\omega)\) is a \(D\)-space for any ordinal number \(\tau,\) where \(\tau_\omega=\{\alpha\leq \tau: \text{cf}(\alpha)\leq \omega\}.\) Furthermore a Lindelöf space of countable pseudocharacter due to Alster, which does not admit a continuous one-to-one mapping onto a first-countable Hausdorff space, is shown to be a \(D\)-space. The author then generalizes known results for \(D\)-spaces (\(\alpha D\)-spaces, respectively). It is shown that a space with a point-countable weak base is a \(D\)-space. As a corollary it follows that a countably compact space with a point-countable weak base is a compact metrizable space. It is also shown that spaces that are a finite union of \(\theta\)-refinable spaces are \(\alpha D\)-spaces.
\(D\)-space, \({\mathcal C}_p(\tau_\omega)\)
scientific article; zbMATH DE number 5123632