On certain generalized versions of groupability
Selection principles in topology have been widely studied in the literature. In this paper, the authors study selection principles in terms of stars, ideals and groupability. Recall that for a subset $A$ of a space $X$ and a family ${\mathcal P}$ of subsets of $X$, the \textit{star of} $A$ with respect to ${\mathcal P}$, denoted by $St(A,{\mathcal P})$, is the set $\bigcup\{P\in{\mathcal P}:A\cap P\neq\emptyset\}$; a family ${\mathcal I}$ of subsets of $X$ is said to be an \textit{ideal} of $X$ if for any $A,B\in{\mathcal I}$, we have that $A\cup B\in{\mathcal I}$, and if $A\in{\mathcal I}$ and $B\subset A$, then $B\in{\mathcal I}$; an element $A$ of a family ${\mathcal A}$ is said to be \textit{groupable} if there is a partition $A=\bigcup_{n\in\mathbb N} A_n$ of $A$ into pairwise disjoint finite sets such that for each infinite subset $N$ of the set of all positive integers, $\bigcup\{A_n:n\in N\}\in{\mathcal A}$, cf. [\textit{Lj. D. R. Kočinac} and \textit{M. Scheepers}, Fundam. Math. 179, No. 2, 131--155 (2003; Zbl 1115.91013)]. Two sample results: \textbf{Theorem 3.2.} For a paracompact space \(X\) if TWO has a winning strategy in the \(S\mathcal{I}H\) game on \(X\) then each \(\mathcal{I}\)-large cover is a star-\(\mathcal{I}\)-groupable as well as an \(\mathcal{I}\)-groupable open cover of \(X\). \textbf{Theorem 4.1.} For a space \(X\) \begin{itemize} \item[(i)] \(SS^*_{fin}(\mathcal{O}, \mathcal{R})\) and \(SS^*_{fin}(\mathcal{O}, \mathcal{O}^{\mathcal{R}-gp})\) are equivalent. \item[(ii)] \(U^*_{fin}(\mathcal{O}, \mathcal{R})\) and \(U^*_{fin}(\mathcal{O}, \mathcal{O}^{\mathcal{R}-gp})\) are equivalent. \item[(iii)] \(^*U_{fin}(\mathcal{O}, \mathcal{R})\) and \(^*U_{fin}(\mathcal{O}, \mathcal{O}^{\mathcal{R}-gp})\) are equivalent. \end{itemize}
scientific article; zbMATH DE number 7047507