Buzyakova, Raushan Z. Hereditary D-property of function spaces over compacta. (English) Zbl 1064.54029 Proc. Am. Math. Soc. 132, No. 11, 3433-3439 (2004). 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. Reviewer: Harry Poppe (Rostock) Cited in 1 ReviewCited in 18 Documents 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 PDFBibTeX XMLCite \textit{R. Z. Buzyakova}, Proc. Am. Math. Soc. 132, No. 11, 3433--3439 (2004; Zbl 1064.54029) Full Text: DOI