Abstract:A {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 {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 Lindel\"of $P$-space is a $D$-space and we prove an addition theorem for metalindel\"of spaces which answers a question of Arhangel'skii and Buzyakova.
Keywords: neighbourhood assignment, $D$-space, dually discrete space, discrete kernel, scattered space, paracompactness, GO-space
AMS Subject Classification: Primary 54D20; Secondary 54G99