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Star-Lindelöf and absolutely star-Lindelöf spaces. (English) Zbl 0931.54019
For a topological space $$X$$ consider the following two properties: For every open cover $${\mathcal U}$$ of $$X$$ there exists a finite (respectively, countable) set $$F\subset X$$ such that $$\text{st}(F,{\mathcal U})= \bigcup\{U\in{\mathcal U}:F\cap U\neq\emptyset\}=X$$. A space with the first property is called star-compact and a space with the second property, which has been introduced under several names, here is called star-Lindelöf. Now consider two stronger properties: For every open cover $${\mathcal U}$$ of $$X$$ and every dense set $$D\subset X$$, there exists a finite (respectively, countable) set $$F\subset D$$ such that $$\text{st}(F,{\mathcal U})=X$$. A space with the first of these properties is called absolutely countably compact. This property was introduced by M. Matveev [Topology Appl. 58, No. 1, 81-92 (1994; Zbl 0801.54021)], and has been studied in some depth. A space with the second property, the main new concept introduced in this paper, is called absolutely star-Lindelöf. The author also considers a number of similar properties using stars and centered families. For example, a space $$X$$ is called centered Lindelöf provided for every open cover $${\mathcal U}$$ there exists a centered family $${\mathcal A}\subset {\mathcal U}$$ such that $$X=\bigcup {\mathcal A}$$. The results in the paper concern relations among spaces with these properties, their images, preimages, products and $$\Sigma$$-products, and cardinal valued functions defined from some of the properties.

##### MSC:
 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54B10 Product spaces in general topology