# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
$D$-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