On linear neighborhood assignments and dually discrete spaces

In this note, the concept of a linear neighborhood assignment is introduced. By discussing properties of linear D-spaces, we show that if T is a Suslin tree with FW (or CW) topology, then T is a Lindelöf D-space. We also show that if X is a countably compact space and X = ⋃ { X n : n ∈ N } , where for any linear neighborhood assignment ϕ n for X n , there exists a strong DC-like subspace (or a subparacompact C-scattered closed subspace) D n of X n , such that X n = ⋃ { ϕ ( d ) : d ∈ D n } for each n ∈ N , then X is a compact space; Every generalized ordered space is dually discrete. This gives a positive answer to a question of Buzyakova, Tkachuk and Wilson.