Set star-Menger and set strongly star-Menger spaces
If \(\mathcal A\) is a family of sets, then the star of a set \(M\subseteq X\) with respect to \(\mathcal A\) is the set \[\textstyle St(M, \mathcal A) =X \cap \bigcup \{U\in \mathcal A \colon M\cap U \not=\emptyset\}.\] A topological space \(X\) is said to have the \textit{set star-Menger property} (the \textit{set strongly star-Menger property}) whenever for each subset \(M\subseteq X\) and any sequence \(\textstyle \{\mathcal{U}_n\colon n \in \mathbb N\}\) of open covers of \(\overline{M} \subseteq X\) there exist finite families \(\mathcal V_n \subseteq \mathcal U_n\) (finite sets \(F_n\subseteq \overline{M} \)) such that \[M \subseteq \bigcup\{St(\bigcup \mathcal V_n, \mathcal U_n)\colon n \in \mathbb N\} \quad (M \subseteq \bigcup\{St(F_n, \mathcal U_n)\colon n \in \mathbb N\}).\] In the reviewed paper notions of the set star-Menger property and the set strongly star-Menger property are introduced. Some examples are given showing relationships between spaces having such properties and other related spaces. Despite the fact that some examples are known in the theory of star selection principles, some additional properties of these are examined. The paper ends with suggestions for several definitions, leaving readers to analyse the properties of such new concepts by analogous methods.
scientific article; zbMATH DE number 7478479
scientific article