D-space explained

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 c up _{x\in} 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]

Notes and References

1. van Douwen. E.. Pfeffer. W.. 1979. Some properties of the Sorgenfrey line and related spaces. Pacific Journal of Mathematics. 81. 2. 371–377. 10.2140/pjm.1979.81.371. free.
2. Book: Elliott., Pearl. Open problems in topology II. 2007-01-01. Elsevier. 9780444522085. 162136062.
3. Aurichi. Leandro. D-Spaces, Topological Games, and Selection Principles. Topology Proceedings. 2010. 36. 107–122.
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. 10.1090/S0002-9939-97-03902-6. 0002-9939. free.