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$D$-spaces and thick covers. (English) Zbl 1244.54056

This paper gives over 15 results relating nearly a dozen subtly defined properties. For readability, this review mentions only a few of these results relating only a few of these properties. The results mentioned in this review all end in “then $X$ is a $D$-space,” but many of the results in this paper do not have this format.

Recall that $X$ is a $D$-space iff, given an open cover $\left\{{U}_{x}:x\in X\right\}$ where each $x\in {U}_{x}$, there is a closed discrete $D\subseteq X$ so that $X={\bigcup }_{x\in D}{U}_{x}$.

To give the flavor of this paper, here are three results:

If $X$ has an almost thick cover by closed Lindelöf $D$-sets, then $X$ is a $D$-space.

Here a cover $ℒ$ is almost thick iff $\forall H\in {\left[X\right]}^{<\omega }$ there is ${L}_{H}$ a finite union of elements of $ℒ$ so that if $A\subset X$ and $A$ is not closed then there is $H\in {\left[A\right]}^{<\omega }$ with ${L}_{H}\cap$ cl$\left(A\setminus A\right)\ne \varnothing$.

If $X$ is $t$-metrizable, then $X$ is a $D$-space.

Here a space $\left(X,\tau \right)$ is $t$-metrizable iff it has a finer metrizable topology $\pi$ and a function $J:{\left[X\right]}^{<\omega }\to {\left[X\right]}^{<\omega }$ so that if $A\subset X$ then cl${}_{\tau }A\subset$ cl${}_{\pi }{\bigcup }_{H\in {\left[A\right]}^{<\omega }}J\left(H\right)$.

If $X$ is a union of finitely many screenable $\sigma$-spaces, then $X$ is a $D$-space.

Here a $\sigma$-space is one with a $\sigma$-discrete closed network; a space is screenable iff each open cover has a $\sigma$-pairwise disjoint open refinement.

##### MSC:
 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54A25 Cardinality properties of topological spaces
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