The D-property which relates to certain covering properties

We show that if X is a regular D ¯ -scattered space and X is the union of a finite collection of submetacompact spaces, then X is a D-space, where D ¯ 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]. second part of this note, we show that the product of a finite collection of regular weak θ ¯ -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 D ¯ -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.