D-space

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In mathematics, a topological space [math]\displaystyle{ X }[/math] is a D-space if for any family [math]\displaystyle{ \{U_x:x\in X\} }[/math] of open sets such that [math]\displaystyle{ x\in U_x }[/math] for all points [math]\displaystyle{ x\in X }[/math], there is a closed discrete subset [math]\displaystyle{ D }[/math] of the space [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ \bigcup_{x\in D}U_x=X }[/math].

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.