In mathematics, a topological space ${\displaystyle X}$ is a D-space if for any family ${\displaystyle \{U_{x}:x\in X\))$ of open sets such that ${\displaystyle x\in U_{x))$ for all points ${\displaystyle x\in X}$, there is a closed discrete subset ${\displaystyle D}$ of the space ${\displaystyle X}$ such that ${\displaystyle \bigcup _{x\in D}U_{x}=X}$.

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]