A note on D-spaces and infinite unions

It is shown that if X is a countably compact space that is the union of a countable family of D-spaces, then X is compact. This gives a positive answer to Arhangel'skii's problem [A.V. Arhangel'skii, D-spaces and finite unions, Proc. Amer. Math. Soc. 132 (7) (2004) 2163–2170]. In this note, we also obtain a result that if a regular space X is sequential and has a point-countable k-network, then X is a D-space.