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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.
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