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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 $X$. {\parindent=4mm \item{--} $X$ is linearly Lindelöf if, and only if, $X$ is a linearly $D$-space of countable extent. \item{--} $X$ is linearly $D$ provided that $X$ is submetaLindelöf. \item{--} $X$ is linearly $D$ provided that $X$ is the union of finitely many linearly $D$-subspaces. \item{--} $X$ is compact provided that $X$ is countably compact and $X$ is the union of countably many linearly $D$-subspaces. \par}

54A25Cardinality properties of topological spaces
54D20Noncompact covering properties (paracompact, Lindelöf, etc.)
Full Text: DOI
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