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A study of $$D$$-spaces. (English) Zbl 0787.54023
A neighborhood assignment for a topological space $$(X,\tau)$$ is a function $$\phi: X \to \tau$$ such that $$x\in \phi(x)$$ for all $$x$$. The following definition is due to van Douwen: $$X$$ is $$D$$-space if, for each neighborhood assignment $$\phi$$, there exists a closed discrete subset $$D_ \phi$$ of $$X$$ such that $$\{\phi(d): d\in D_ \phi\}$$ covers $$X$$. In this paper, the authors present several fundamental results regarding $$D$$-spaces. They show that $$D$$-spaces are preserved by closed images and by perfect preimages. They show that paracompact $$p$$-spaces are $$D$$- spaces, that monotonically normal $$D$$-spaces are paracompact, and that semistratifiable spaces are $$D$$-spaces [this last result was announced by P. de Caux in [Topology, Proc. Conf., Vol. 6, No. 1, Blacksburg/Va. 1981, 31-43 (1982; Zbl 0535.54008)], where he also showed that subspaces of finite products of the Sorgenfrey line are $$D$$-spaces, a question raised here].

##### MSC:
 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54E20 Stratifiable spaces, cosmic spaces, etc. 54E18 $$p$$-spaces, $$M$$-spaces, $$\sigma$$-spaces, etc.
##### Keywords:
$$D$$-space; neighborhood assignment
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