Some topological minimax theorems (Q1581467)

Let \(X, Y\) be topological spaces and \(F(x,y)\) be defined on \(X \times Y\). The minimax theorems establish conditions under which \[ \inf_{y \in Y} \sup_{x \in X}F(x,y)= \sup_{x \in X} \inf_{y \in Y}F(x,y). \] Two such theorems are proved with no restrictions on \(X,Y\) except for those related to the function \(F(x,y)\). The conditions concerning the behavior of the function deal only with the interval between the minimax and maximin. The scheme of arguments goes back to the Hahn-Banach theorem and the separating hyperplane theorem.