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.