The D-property which relates to certain covering properties

We show that if X is a regular -scattered space and X is the union of a finite collection of submetacompact spaces, then X is a D-space, where is the class of all D-spaces, hence X is a D-space if X is the union of a finite collection of submetacompact C-scattered spaces, where C is the class of all compact spaces. This gives a positive answer to a question mentioned by Martínez [J.C. Martínez, On finite unions and finite products with the D-property, Topology Appl. 158 (2) (2011) 223–228].In the second part of this note, we show that the product of a finite collection of regular weak -refinable (or (sub)metacompact) C-scattered spaces and a Lindelöf D-space is a D-space. In the last part of this note, we show that if X is a regular meta-Lindelöf -scattered space with locally countable extent, then X is a D-space. As a corollary, every meta-Lindelöf locally compact Hausdorff space is a D-space.