The D-property which relates to certain covering properties

摘要

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 Martinez [J.C. Martinez, 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 ft-refinable (or (sub)metacompact) C-scattered spaces and a Lindelof D-space is a D-space. In the last part of this note, we show that if X is a regular meta-Lindelof -D-scattered space with locally countable extent, then X is a D-space. As a corollary, every meta-Lindelof locally compact Hausdorff space is a D-space.