On (strong) \(\alpha \)-favorability of the Wijsman hyperspace (Q712210)

Given a metric space \((X,d)\), let \(CL(X)\) denote the hyperspace of nonempty closed subsets of \(X\) endowed with the Wijsman topology. The authors investigate the Banach-Mazur and the strong Choquet game on \(CL(X)\) from the nonempty player's (i.e., \(\alpha \)'s or second player) perspective. Since the strategies for these games are related to some completeness properties of the spaces, through their study it is possible to explore completeness properties of the hyperspaces \(CL(X)\). In this paper, for the strong Choquet game it is shown that if \(X\) is a locally separable metrizable space, then \(\alpha \) has a (stationary) winning strategy on \(X\) iff it has a (stationary) winning strategy on \(CL(X)\) for each compatible metric on \(X\). The analogous result for the Banach-Mazur game does not hold, not even if \(X\) is separable, as it is shown that \(\alpha \) may have a (stationary) winning strategy on \(CL(X)\) for each compatible metric on \(X\), and may not have one on \(X\). It is also shown than there exists a separable 1st category metric space such that \(\alpha \) has a (stationary) winning strategy on \(CL(X)\). This answers Question 4.2 of \textit{J. Cao} and \textit{H. J. K. Junnila} [Proc. Am. Math. Soc. 138, No.~2, 769--776 (2010; Zbl 1192.54011)]. The paper ends with the following open problem: Does there exist a metrizable non-Baire space \(X\) such that \(CL(X)\) is a Baire space for each compatible metric on \(X\)?