Limited information strategies and discrete selectivity (Q2324526)

Given a space \(X\), the game \(CD(X)\) is played by two opponents \(O\) and \(P\). In the \(n\)-th round Player \(O\) takes a non-empty open subset \(U_n\) of the space \(X\) and \(P\) responds by choosing a point \(x_n\in U_n\). After \(\omega\)-many moves are made, Player \(P\) is the winner of the game \(\mathcal{L}=\{U_n, x_n: n\in\omega\}\) if the set \(S(\mathcal{L})=\{x_n: n\in\omega\}\) is closed and discrete in \(X\). Otherwise \(O\) is the winner. A family \(\mathcal A\) of subsets of \(X\) is an \textit{\(\omega\)-cover} of \(X\) if for any finite set \(K\subset X\), there exists \(A\in \mathcal A\) such that \(K\subset A\). The game \(PO_\omega(X)\) also has \(\omega\)-many rounds; it is played on a space \(X\) by two opponents called \(O\) and \(P\). In the \(n\)-th round Player \(P\) takes a point \(x_n\in X\) and \(O\) responds by picking an open set \(U_n\) such that \(x_n\in U_n\). After \(\omega\)-many moves are made, the family \(\{x_n, U_n: n\in \omega\}\) is called the \textit{play} of the game and Player \(O\) wins if the family \(\{U_n : n\in \omega\}\) is not an \(\omega\)-cover of \(X\). Otherwise, the victory is assigned to Player \(P\). The main result of the paper states that, for any Tychonoff space \(X\), Player \(O\) has a winning strategy in the game \(PO_\omega(X)\) if and only if Player \(P\) has a winning strategy in the game \(CD(C_p(X))\).