# 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. (English) Zbl 0983.54024
For a space $(X,\tau)$, a function $U:X\to\tau$ such that each $x\in U(x)$ is called an open neighborhood assignment (ONA). $(X,\tau)$ is a $D$-space if for each ONA, $U$, there exists a closed discrete $D\subset X$ such that $\bigcup\{U(x) \mid x\in D\}= U(D)=X$. For an ONA $U:x\to \tau$ and $D \subset X$, $D$ is said to be $U$-sticky if $D$ is closed discrete and $x\in U(D)$ whenever $U(x)\cap D\ne \emptyset$. 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
Full Text:
##### References:
 [1] Borges, C.; Wehrley, A.: A study of D-spaces. Topology proc. 16, 7-15 (1991) [2] Borges, C.; Wehrley, A.: Another study of D-spaces. Questions answers topology 14, 73-76 (1996) [3] Decaux, P.: Yet another property of the sorgenfrey plane. Topology proc. 6, 31-43 (1981) [4] Creede, G.: Concerning semistratifiable spaces. Pacific J. Math. 32, 47-54 (1970) · Zbl 0189.23304 [5] Van Douwen, E.; Lutzer, D.: A note on paracompactness in generalized ordered spaces. Proc. amer. Math. soc. 125, 1237-1245 (1997) · Zbl 0885.54023 [6] Kunen, K.: Set theory. (1980) · Zbl 0443.03021 [7] Lutzer, D.: Ordered topological spaces, surveys. General topology, 247-295 (1980) [8] A. Stanley, D-spaces and a Dowker space, Thesis, Univ. Kansas, 1997 [9] Kemoto, N.; Tamano, K.; Yajima, Y.: Generalized paracompactness of subspaces in products of two ordinals. Topology appl. 104, 155-168 (2000) · Zbl 0945.54017