D-space

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

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. 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.

Properties

• Every Menger space is a D-space.
• A subspace of a topological linearly ordered space is a D-space iff it is a paracompact space.