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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 of topological spaces 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)