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 D -space introduced by E. van Douwen and on a generalization of this notion, the notion of a D aD -space. It is proved that if a space X X is the union of a finite family of subparacompact subspaces, then X X is an a D aD -space. Under ( C H ) (CH) , it follows that if a separable normal T 1 T_1 -space X X is the union of a finite number of subparacompact subspaces, then X X is Lindelöf. It is also established that if a regular space X X is the union of a finite family of subspaces with a point-countable base, then X X is a D 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.