Topological properties defined by games and their applications
The authors investigate two-person games, called \({\mathcal G}({\mathcal F})\)-games, concerning a filter (filterbase) \({\mathcal F}\) in a space \(X\), and study topological properties defined by them. For a filter (filterbase) \({\mathcal F}\) in a space \(X\), the \({\mathcal G}({\mathcal F})\)-game played in \(X\) between players \(A\) and \(B\) is defined as follows: First, player \(A\) chooses a point \(x_1\in X\), and player \(B\) must then respond by choosing a member \(F_1\in{\mathcal F}\). Next, player \(A\) must select another (possibly the same) point \(x_2\in F_1\) and in turn player \(B\) must again respond to this by choosing a member \(F_2\in{\mathcal F}\). Repeating this infinitely, \(A\) and \(B\) produce a sequence \(\{(x_n,F_n):n\in \mathbb{N}\}\) with \(x_{n+1}\in F_n\) for each \(n\in \mathbb{N}\), called a play of the \({\mathcal G}({\mathcal F})\)-game. Then, \(B\) is said to have won this play if the sequence \(\{x_n\}\) has a cluster point in \(X\), and otherwise, \(A\) is said to have won this play. A pair \(({\mathcal F},\sigma)\) of a proper filter (filterbase) \({\mathcal F}\) in a space \(X\) and a winning strategy \(\sigma\) in the \({\mathcal G}({\mathcal F})\)-game is called a \(\sigma\)-filter (\(\sigma\)-filterbase). A space \(X\) is said to have property \((\ast\ast)\) if \(\bigcap\{\overline{F}:F\in{\mathcal F}\}\neq\emptyset\) for each \(\sigma\)-filterbase \(({\mathcal F},\sigma)\) in \(X\). The main results are: (i) If \(({\mathcal F},\sigma)\) is a \(\sigma\)-filterbase in a regular space with property \((\ast\ast)\), then the set \(K=\bigcap\{\overline{F}:F\in{\mathcal F}\}\) is compact and every open set containing \(K\) contains some member of \({\mathcal F}\). (ii) Property \((\ast\ast)\) is productive. (iii) Every Dieudonné-complete space has property \((\ast\ast)\). (iv) The space of real-valued continuous functions on a compact Hausdorff space with the pointwise convergence topology has property \((\ast\ast)\). (i) enables the authors to improve theorems by \textit{S. Dolecki} and \textit{A. Lechicki} [J. Math. Anal. Appl. 88, 547-554 (1982; Zbl 0503.54023)] and by \textit{R. W. Hansell, J. E. Jayne, I. Labuda} and \textit{C. A. Rogers} [Math. Z. 189, 297-318 (1985; Zbl 0544.54016)] on multi-valued functions. For a point \(x\) of a space \(X\), let \({\mathcal U}(x)\) denote the neighborhood system of \(x\). A regular point is a point \(x\) such that each member of \({\mathcal U}(x)\) contains the closure of some member of \({\mathcal U}(x)\). A point \(x\) is called a \({\mathcal G}(x)\)-point if there exists a winning strategy \(\sigma\) such that \(({\mathcal U}(x),\sigma)\) is a \(\sigma\)-filter. A space \(X\) is called a \({\mathcal G}\)-space if each point \(x\in X\) is a \({\mathcal G}(x)\)-point. Another application of (i) is the following generalization of classical Vajnshtejn's Lemma: Let \(f\) be a closed continuous map from a space \(X\) with property \((\ast\ast)\) onto a Hausdorff \({\mathcal G}\)-space \(T\). Then the boundary of \(f^{-1}(t)\) is compact for each \(t\in T\).
scientific article; zbMATH DE number 1801749