A note on D-spaces

We show that every regular T1 submeta-Lindelöf space of cardinality ω1 is D under MA+¬CH, which answers a question posed by Gruenhage (2011) [9], . Borges (1991) [5] asked if every monotonically normal paracompact space is a D-space, we give a characterization of paracompactness for monotonically normal spaces, which may be of some use in solving this problem.