Predictable network, monotonic monolithicity and \(D\)-spaces
\(D\)-spaces were introduced by \textit{E. K. van Douwen} and \textit{W. F. Pfeffer} in the paper [Pac. J. Math. 81, 371--377 (1979; Zbl 0409.54011)]. A space \(X\) is called a \(D\)-space if for every mapping \(\phi\) from \(X\) to its topology \(\tau\) with \(x\in \phi(x)\) for every \(x\in X\), there is a closed and discrete subset \(A\) of \(X\) such that \(\phi(A)=\{\phi(x): x\in A\}\) covers \(X\). The famous problem whether a regular Lindelöf or paracompact space is \(D\) has attracted wide attention and study and much intensive work has been done, cf. [\textit{G. Gruenhage}, Contemp. Math. 533, 13--28 (2011; Zbl 1217.54025)]. In the paper under review, the authors study different concepts related \(D\)-spaces. They mainly show the following: (1) any space with an \(\omega_0\)-Noetherian point network (a point network is also called ``condition (F)'' or ``Collins-Roscoe structuring mechanism'') is hereditarily thickly covered; (2) there is a monotonically monolithic space which doesn't have a predictable network under the assumption \(2^{\omega_0}<2^{\omega_1}\), and this provides a consistent negative answer to one of \textit{H. Guo} and \textit{H. Junnila}'s questions in the paper [Topology Appl. 157, No. 1, 102--107 (2010; Zbl 1180.54009)]; (3) a strong monotonically monolithic space \(X\) has a point-countable base if and only if \(X\) has a dense, point-countable, pointed open cover \(\mathcal{P}\) (that is, a set \(\mathcal{P}\) of pairs \((x, U)\) with \(x\in X\) and \(U\) open such that \(\{(x, U)\in \mathcal{P}: y\in U\}\) is countable for all \(y\in X\), and \(y \in \overline{\{x: y\in U\wedge (x, U)\in \mathcal{P}\}}\)); (4) any strong monotonically monolithic space which is semi-stratifiable and metaLindelöf has a point-countable base. The authors also pose the following problem: is a space \(D\) if it is a finite union of monotonically monolithic subspaces?
(strong) monotonically monolithic spaces
scientific article; zbMATH DE number 7005609
scientific article