Star-Lindelöf and absolutely star-Lindelöf spaces.

*(English)*Zbl 0931.54019For 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.

Reviewer: Jerry E.Vaughan (Greensboro)