A note on D-spaces and L-special trees

In this note, we show that if L is a totally ordered set such that there exists a countable set C ⊂ L satisfying that ( a , b ) ∩ C ≠ ∅ whenever a , b ∈ L and ( a , b ) ≠ ∅ , then any L-special tree T is hereditarily a D-space. This generalizes some conclusions of H.C. Gamel. We prove that if L 1 and L 2 are Dedekind complete totally ordered sets such that all L 1 -special trees and all L 2 -special trees are hereditarily D-spaces, where L 1 × L 2 is under the lexicographic ordering, then any L 1 × L 2 -special tree is hereditarily a D-space. By this conclusion and a conclusion of H.C. Gamel, we show that if α < ω 2 and [ 0 , 1 ] α is under the lexicographic ordering, then all [ 0 , 1 ] α -special trees are hereditarily D-spaces. Thus some results of H.C. Gamel are generalized. ∈ ω 1 be a limit ordinal and let I α be a Dedekind complete linearly ordered metric space for each α ∈ β such that max I α and min I α exist. If L = ∏ α ∈ β I α is under the lexicographic ordering such that any L α -special tree is hereditarily a D-space for each α ∈ β , where L α = ∏ γ ∈ α I γ is under the lexicographic ordering, then every L-special tree T satisfies that the height of each branch of T is countable. As a corollary, we get that if T is a [ 0 , 1 ] ω 2 -special tree, where [ 0 , 1 ] ω 2 is under the lexicographic ordering, then the height of each branch of T is countable.