Selection games on hyperspaces (Q2049887)

In this paper, the authors unify techniques and generalize theorems from the known results about selection principles for common hyperspace constructions. This includes results of Lj. D. R. Kǒcinac, Z. Li, and others. They use selection games to generalize selection principles and work with strategies of various strengths for these games. The main results are the following: \textbf{Theorem 1.} Suppose that \(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}\) are collections so that \(\bigcup\mathcal{C}\subset\bigcup\mathcal{A}\) and \(\bigcup\mathcal{D}\subset\bigcup\mathcal{B}\). Additionally, suppose that there exists a bijection \(\beta:\bigcup\mathcal{A}\rightarrow\bigcup\mathcal{B}\) with the following features: \begin{itemize} \item \(A\in\mathcal{A}\Leftrightarrow\beta[A]\in\mathcal{B}\); \item \(C\in\mathcal{C}\Leftrightarrow\beta[C]\in\mathcal{D}\). \end{itemize} Then the selection game defined by \(\mathcal{A}\) and \(\mathcal{C}\), and the game defined by \(\mathcal{B}\) and \(\mathcal{D}\) are equivalent. This equivalence holds whether the games use single selections or finite selections. \textbf{Theorem 2.} Fix a topological space \(X\). Then \begin{itemize} \item[(i)] The generalized Rothberger/Menger game on \(X\) is equivalent to the Selective Separability game played on the generalized upper Fell topology; \item[(ii)] The generalized local Rothberger/Menger game on \(X\) is equivalent to the Countable Fan Tightness game played on the generalized upper Fell topology, and \item[(iii)] The generalized Hurewicz game on \(X\) is equivalent to a modified Selective Separability game played on the generalized upper Fell topology. \end{itemize} \textbf{Theorem 3.} The previous theorem holds for \(X\) and the generalized full Fell topology if the covers of \(X\) are replaced by \(\mathcal{A}_{F}\)-covers.