D-space

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In mathematics, a D-space is a topological space where for every neighborhood assignment of that space, a cover can be created from the union of neighborhoods from the neighborhood assignment of some closed discrete subset of the space.

Definition

An open neighborhood assignment is a function that assigns an open neighborhood to each element in the set. More formally, given a topological space $X$ . An open neighborhood assignment is a function $f: X \to N(X)$ where $f(x)$ is an open neighborhood.

A topological space $X$ is a D-space if for every given neighborhood assignment $N_x : X \to N(X)$ , there exists a closed discrete subset $D$ of the space $X$ such that $\bigcup_{x\in D}N_x=X$ .

History

The notion of D-spaces was introduced by Eric Karel van Douwen and E.A. Michael. It first appeared in a 1979 paper by van Douwen and Washek Frantisek Pfeffer in the Pacific Journal of Mathematics.{{Cite journal|last1=van Douwen|first1=E.|last2=Pfeffer|first2=W.|year=1979|title=Some properties of the Sorgenfrey line and related spaces|url=http://msp.org/pjm/1979/81-2/pjm-v81-n2-p07-s.pdf|journal=Pacific Journal of Mathematics|volume=81|issue=2|pages=371–377|doi=10.2140/pjm.1979.81.371|doi-access=free}} Whether every Lindelöf and regular topological space is a D-space is known as the D-space problem. This problem is among twenty of the most important problems of set theoretic topology.{{Cite book|url=https://www.worldcat.org/oclc/162136062|title=Open problems in topology II|last=Elliott.|first=Pearl|date=2007-01-01|publisher=Elsevier|isbn=9780444522085|oclc=162136062}}

Properties

Every Menger space is a D-space.{{cite journal|last1=Aurichi|first1=Leandro|title=D-Spaces, Topological Games, and Selection Principles|

journal=Topology Proceedings|date=2010|volume=36|pages=107–122|url=http://topology.auburn.edu/tp/reprints/v36/tp36009.pdf}}

A subspace of a topological linearly ordered space is a D-space iff it is a paracompact space.{{Cite journal|last1=van Douwen|first1=Eric|last2=Lutzer|first2=David|date=1997-01-01|title=A note on paracompactness in generalized ordered spaces|url=https://www.ams.org/proc/1997-125-04/S0002-9939-97-03902-6/|journal=Proceedings of the American Mathematical Society|volume=125|issue=4|pages=1237–1245|doi=10.1090/S0002-9939-97-03902-6|issn=0002-9939|doi-access=free}}

References

Category:Properties of topological spaces

Category:Topology