On finite unions of certain D-spaces

In this note, we show that if X is the union of a finite collection { X i : i = 1 , … , k } of strong Σ-spaces, then X is a D-space. As a corollary, we get a conclusion that if X is the union of a finite collection { X i : i = 1 , … , k } of Moore spaces, then X is a D-space. This gives a positive answer to one of Arhangelʹskiiʹs problems [A.V. Arhangelʹskii, D-spaces and finite unions, Proc. AMS 132 (7) (2004) 2163–2170]. In the last part of the note, we show that if X is the union of a finite collection { X i : i = 1 , … , k } of DC-like spaces, then X is a D-space, where DC is the class of all discrete unions of compact spaces. As a corollary, we show that if X is the union of a finite collection of regular subparacompact C-scattered spaces, then X is a D-space.