A definitive improvement of a game-theoretic bound and the long tightness game (Q1669676)

For an ordinal \(\kappa\), families \(\mathcal A\) and \(\mathcal B\) and \(\nu\), \({\mathbb G}_\nu^\kappa(\mathcal A,\mathcal B)\) denotes the game played by two players such that at inning \(\xi<\kappa\) the first player chooses \(A_\xi\in\mathcal A\) then the second player chooses \(S_\xi\subset\mathcal A_\xi\): the second player wins if \(\cup_{\xi<\kappa}S_\xi\in\mathcal B\). If \(\nu=1\), resp. 2, we demand that \(S_\xi\) is a singleton, resp a doubleton, finite. A regular space with points \(G_\delta\) in which the second player has a winning strategy in \({\mathbb G}_{\text fin}^{\omega_1}(\mathbb O,\mathbb O)\) has cardinality at most \(\mathbb C\), where \(\mathbb O\) is the family of all open covers of \(X\). An example is given of a \(T_1\) space with points \(G_\delta\) and cardinality at least \(\kappa\;(<\) the first measurable cardinal) in which the second player has a winning strategy in \({\mathbb G}_{\text fin}(\mathbb O,\mathbb O)\). The game \({\mathbb G}_{\nu}^{\omega_1}(\Omega_p,\Omega_p)\), where \(\Omega_p\) is the collection of all subsets \(A\subset X\) satisfying \(p\in\overline{A}\), is also studied. An example is given of a zero-dimensional \(T_1\) space in which the first player has a winning strategy in \({\mathbb G}_{1}^{\omega_1}(\Omega_p,\Omega_p)\) but the second player has a winning strategy in \({\mathbb G}_{2}^{\omega_1}(\Omega_p,\Omega_p)\).