On spaces which are D, linearly D and transitively D

In this note, we comment on D-spaces, linearly D-spaces and transitively D-spaces. We show that every meta-Lindelöf space is transitively D. If X is a weak θ ¯ -refinable TD -scattered space, then X is transitively D, where TD is the class of all transitively D-spaces. If X is a weak θ ¯ -refinable D ¯ -scattered space, then X is a D-space, where D ¯ is the class of all D-spaces, and hence every weak θ ¯ -refinable (or submetacompact) scattered space is a D-space. This gives a positive answer to a question mentioned by Martínez and Soukup. In the last part of this note, we show that if X is a weak θ ¯ -refinable space then X is linearly D.