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Hereditary D-property of function spaces over compacta. (English) Zbl 1064.54029

Eric K. van Douwen [Thesis, Free University of Amsterdam (1975)] introduced the notion of a D-space: Let \((X,\tau)\) be a topological space; a neighborhood assignment for X is a function \(\phi\) from \(X\) to \(\tau\) such that \(x\in\phi(x)\) for all \(x\in X\). \(X\) is a D-space, if for any neighborhood assignment \(\phi\) for \(X\) there exists a closed discrete subset \(D\) of \(X\) such that \(X=\bigcup_{d\in D}\phi(d)\).
The author shows the interesting theorem that for a compact Hausdorff topological space \(X\) each subset \(Y\subseteq C_p(X)\) is a D-space. As it is pointed out at the end of the paper (after submission remarks), G. Gruenhage showed that the theorem already holds for \(\Sigma\)-Lindelöf spaces. The author also generalizes his theorem by showing that if \(X\) belongs to a special class of countably compact spaces then the theorem holds, too.

MSC:

54C35 Function spaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54C60 Set-valued maps in general topology

Keywords:

\(C_p(X)\); D-space
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