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D-spaces. (English) Zbl 0983.54024

For a space $\left(X,\tau \right)$, a function $U:X\to \tau$ such that each $x\in U\left(x\right)$ is called an open neighborhood assignment (ONA). $\left(X,\tau \right)$ is a $D$-space if for each ONA, $U$, there exists a closed discrete $D\subset X$ such that $\bigcup \left\{U\left(x\right)\mid x\in D\right\}=U\left(D\right)=X$. For an ONA $U:x\to \tau$ and $D\subset X$, $D$ is said to be $U$-sticky if $D$ is closed discrete and $x\in U\left(D\right)$ whenever $U\left(x\right)\cap D\ne \varnothing$. Among other results, it is proved that

(1) Box products of scattered spaces of height 1 are $D$-spaces,

(2) A subspace of a linearly ordered space is a $D$-space iff it has no closed stationary subset (a harder proof of this result is due to van Douwen),

(3) A subspace of the product of finitely many ordinals is a $D$-space iff it is metacompact iff it has no closed stationary subsets.

##### MSC:
 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54E20 Stratifiable spaces, cosmic spaces, etc. 54F05 Linearly, generalized, and partial ordered topological spaces
##### Keywords:
open neighborhood assignment; scattered spaces