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

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].