A topological KKM theorem and minimax theorems
The Knaster-Kuratowski-Mazurkiewicz theorem for a class of multifunctions between 2 Hausdorff topological spaces (without linear structure) \(X\) and \(Y\) with the following property: \[ \underset {x_0 \in X} \forall\;\underset {\{x_n\}, x_n \to x_0} \forall\;[\text{ if } y_0 \notin F(x_n) \text{ for all \(n\), then } y_0 \notin F(x_0)] \] is proved; here \(\{x_n\}\) is a net in \(X\). This theorem is then used in the proof of the following ``minimax theorem'': If \(f, g : X \times Y \to \mathbb{R}\) are functions such that: \(g(x, \cdot)\), \(g(\cdot, y)\) both are u.s.c., \(f(x, \cdot)\) is l.s.c., \(f(x,y) \leq g(x,y)\) for all \(x\in X\), \(y \in Y\), for any finite set \(\{x_1, \dots, x_n\} \subset X\) and any \(r\in \mathbb{R}\) the set \(\{y \in Y : g(x_i,y) < r\), \(i = 1, \dots, n\}\) is connected, for any \(x_1, x_2 \in X\) there exists a connected subset \(X \supset C \supset \{x_1, x_2\}\) such that \(g(x,y) \geq \max \{g(x_1, y), g(x_2, y)\}\) for all \(x \in C\) and all \(y \in Y\), \(Y\) is a compact space, then \(\sup_{x \in X} \inf_{y \in Y} g(x,y) \geq \inf_{y \in Y} \sup_{x \in X} f(x,y).\) Four slightly different versions of this ``minimax theorem'' are also proved.
scientific article; zbMATH DE number 584088