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 X {\displaystyle X} . An open neighborhood assignment is a function f : X → N ( X ) {\displaystyle f:X\to N(X)} where f ( x ) {\displaystyle f(x)} is an open neighborhood. A topological space X {\displaystyle X} is a D-space if for every given neighborhood assignment N x : X → N ( X ) {\displaystyle N_{x}:X\to N(X)} , there exists a closed discrete subset D {\displaystyle D} of the space X {\displaystyle X} such that ⋃ x ∈ D N x = X {\displaystyle \bigcup _{x\in D}N_{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.
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