The D-property of monotone covering properties and related conclusions

In this note, we show that every monotonically (countably) metacompact space is hereditarily a D-space and every monotonically meta-Lindelöf space is hereditarily dually σ-closed discrete. As a corollary, we show that if X is a monotonically meta-Lindelöf (or monotonically (countably) metacompact) monotonically normal space then X is hereditarily paracompact. In the second part of this note, we show that every scattered partition of a hereditarily almost thickly covered space is almost thick, and hence a hereditarily almost thickly covered space is aD and linearly D. This answers a question of Guo and Junnila. We also show that every monotonically ω-monolithic compact space is monotonically monolithic. This answers a question of Alas, Tkachuk and Wilson.