\(D\)-spaces and covering properties
A neighbourhood assignment on a topological space is a mapping \(\phi\) of \(X\) into the topology \(\tau\) of \(X\) such that \(x\in \phi(x)\) for each \(x \in X\). A space \(X\) is a D-space if, for every neighbourhood assignment \(\phi\) on \(X\), there exists a locally finite (in \(X\)) subset \(A\) of \(X\) such that the family \(\phi(A)\) covers \(X\). This paper is a continuation of the author's earlier paper (with \textit{R. Z. Buzyakova}) [Commentat. Math. Univ. Carol. 43, 653--663 (2002)]. We quote the author's abstract: We study the D-space property and its generalizations, the notions of an aD-space and a weak aD-space in connection with covering properties. A brief survey on D-spaces is presented in Section 1. Among new results, it is proved that if a linearly ordered space is an aD-space, then it is paracompact. The statement further extends the list of equivalences in [\textit{E. K. van Douwen} and \textit{D. J. Lutzer}, Proc. Am. Math. Soc. 125, 1237--1245 (1997; Zbl 0885.54023)]. We also establish some sufficient conditions for the free topological group of a Tychonoff space to be a D-space. In particular, the free topological group of a semi-stratifiable space is shown to be a D-space, while it need not be semi-stratifiable. A similar result is established for the free topological group of a space with a point-countable base. Some new interesting open problems on D-spaces and on spaces close to them are formulated. In particular, we discuss several such questions in connection with the sum theorems.
scientific article; zbMATH DE number 2136482
scientific article