D-spaces and covering properties

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 Proc. Amer. Math. Soc. 125 (1997) 1237]. 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.