Some sufficiency conditions for D-spaces, dually discrete and its applications

Abstract

In this note, we obtain some sufficiency conditions for D-spaces, dually discrete, and dually scattered of rank≤2. As an application, we show that if a space X has a chain (F) point network W={W(x):x∈X} such that W(x) is a chain and is a closure-preserving family of subsets of X for each x∈X then X is a paracompact D-space. If a space X has an ω1-sheltering chain (F) point network W={W(x):x∈X} such that W(x) is a chain and is a weak ω-closure-preserving family of subsets of X for each x∈X, then X is a paracompact D-space.If X is a chain neighborhood (F) space, then for each neighborhood assignment ϕ for X there is an open neighborhood Vx of x for each x∈X such that x∈Vx⊂ϕ(x) and for each closed discrete subspace F1 of X and for each closed subset A of X, there exists a closed discrete subspace D of X such that D⊂A∩{z∈X∖ϕ(F1):Vz∩F1≠∅} and A∩{z∈X∖ϕ(F1):Vz∩F1≠∅}⊂ϕ(D). By the above conclusion and a sufficiency condition for D-spaces in this note, we can get a known conclusion that every chain neighborhood (F) space is a D-space [8].