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]