D-spaces and thick covers

The following results are obtained.–An open neighbornet U of X has a closed discrete kernel if X has an almost thick cover by countably U-close sets.–Every hereditarily thickly covered space is aD and linearly D.–Every t-metrizable space is a D-space.–X is a D-space if X has a cover {Xα:α<λ} by D-subspaces such that, for each β<λ, the set ⋃{Xα:α<β} is closed.