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Keywords:
neighbourhood assignment; $D$-space; dually discrete space; discrete kernel; scattered space; paracompactness; GO-space
neighbourhood assignment; $D$-space; dually discrete space; discrete kernel; scattered space; paracompactness; GO-space
Summary:
A {\it neighbourhood assignment\/} in a space $X$ is a family $\Cal O= \{O_x:x\in X\}$ of open subsets of $X$ such that $x\in O_x$ for any $x\in X$. A set $Y\subseteq X$ is {\it a kernel of $\Cal O$\/} if $\Cal O(Y)=\bigcup\{O_x:x\in Y\}=X$. If every neighbourhood assignment in $X$ has a closed and discrete (respectively, discrete) kernel, then $X$ is said to be a $D$-space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf $P$-space is a $D$-space and we prove an addition theorem for metalindelöf spaces which answers a question of Arhangel'skii and Buzyakova.
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