LAUSR

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 ω 1 -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 V x of x for each x ∈ X such that x ∈ V x ⊂ ϕ ( x ) and for each closed discrete subspace F 1 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 ( F 1 ) : V z ∩ F 1 ≠ ∅ } and A ∩ { z ∈ X ∖ ϕ ( F 1 ) : V z ∩ F 1 ≠ ∅ } ⊂ 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] .