## A solution of Matveev’s question.(English)Zbl 1009.54028

Summary: A space $$X$$ is discretely absolutely star-Lindelöf if for every open cover $${\mathcal U}$$ of $$X$$ and every dense subset $$D$$ of $$X$$, there exists a countable subset $$F\subseteq D$$ such that $$F$$ is discrete closed in $$X$$ and $$St(F,{\mathcal U})=X$$. We construct an example of a normal discretely absolutely star-Lindelöf space with an uncountable discrete closed subspace under $$2^{\aleph_0} =2^{\aleph_1}$$ which gives a partial answer to a question of M. V. Matveev.

### MSC:

 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54B10 Product spaces in general topology 54D55 Sequential spaces

### Keywords:

property (a); discretely star-Lindelöf