D-space

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 ${\displaystyle X}$. An open neighborhood assignment is a function ${\displaystyle f:X\to N(X)}$ where ${\displaystyle f(x)}$ is an open neighborhood.

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

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. ^[1] 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. ^[2]

Properties

• Every Menger space is a D-space. ^[3]
• A subspace of a topological linearly ordered space is a D-space iff it is a paracompact space. ^[4]

References

1. van Douwen, E.; Pfeffer, W. (1979). "Some properties of the Sorgenfrey line and related spaces" (PDF). Pacific Journal of Mathematics. 81 (2): 371–377. doi: 10.2140/pjm.1979.81.371 .
2. Elliott., Pearl (2007-01-01). Open problems in topology II. Elsevier. ISBN   9780444522085. OCLC   162136062.
3. Aurichi, Leandro (2010). "D-Spaces, Topological Games, and Selection Principles" (PDF). Topology Proceedings. 36: 107–122. Archived from the original (PDF) on 2017-01-09. Retrieved 2017-01-24.
4. van Douwen, Eric; Lutzer, David (1997-01-01). "A note on paracompactness in generalized ordered spaces". Proceedings of the American Mathematical Society. 125 (4): 1237–1245. doi: 10.1090/S0002-9939-97-03902-6 . ISSN   0002-9939.