$D$-spaces and finite unions

Abstract:

This article is a continuation of a recent paper by the author and R. Z. Buzyakova. New results are obtained in the direction of the next natural question: how complex can a space be that is the union of two (of a finite family) “nice" subspaces? Our approach is based on the notion of a $D$-space introduced by E. van Douwen and on a generalization of this notion, the notion of $aD$-space. It is proved that if a space $X$ is the union of a finite family of subparacompact subspaces, then $X$ is an $aD$-space. Under $(CH)$, it follows that if a separable normal $T_1$-space $X$ is the union of a finite number of subparacompact subspaces, then $X$ is Lindelöf. It is also established that if a regular space $X$ is the union of a finite family of subspaces with a point-countable base, then $X$ is a $D$-space. Finally, a certain structure theorem for unions of finite families of spaces with a point-countable base is established, and numerous corollaries are derived from it. Also, many new open problems are formulated.