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. An open neighborhood assignment is a function where is an open neighborhood.A topological space is a D-space if for every given neighborhood assignment, there exists a closed discrete subset of the space such that.
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.