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On spaces which are linearly \(D\). (English) Zbl 1180.54009
Summary: We introduce a generalization of \(D\)-spaces, which we call linearly \(D\)-spaces. The following results are obtained for a \(T_{1}\)-space \(X\).
\(X\) is linearly Lindelöf if, and only if, \(X\) is a linearly \(D\)-space of countable extent.
\(X\) is linearly \(D\) provided that \(X\) is submetaLindelöf.
\(X\) is linearly \(D\) provided that \(X\) is the union of finitely many linearly \(D\)-subspaces.
\(X\) is compact provided that \(X\) is countably compact and \(X\) is the union of countably many linearly \(D\)-subspaces.

MSC:
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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