A Dini-Dax theorem
Let \(X\) be a nonvoid set, \(Y\) a compact topological space, and \(f:X\times Y\to \mathbb{R}\) be a real function, lower-semicontinuous in the second variable. \textit{H. König} [Arch. Math. 19, 482-487 (1968; Zbl 0179.21001)] gave a version of the well-known Dini-theorem: The ``Dini-condition'' \[ \forall x_1, x_2\in X\;\exists x_0\in X\;\forall y\in Y:f(x_0,y) \geq\max \bigl( f(x_1,y), f(x_2,y)\bigr) \] implies \(\inf_{y\in Y}\sup_{x\in X} f(x,y)= \sup_{x\in X} \inf_{y\in Y} f(x,y)\). In 1977, H. König introduced a ``counterpart'' of the Dini-theorem, called Dax-theorem: The ``Dax-condition'' \[ \forall y_1, y_2\in Y\;\exists y_0\in Y\;\forall x\in X:f(x,y_0) \leq\min \bigl( f(x, y_1), f(x,y_2) \bigr) \] implies \(\inf_{y\in Y} \sup_{x\in X} f(x,y)= \sup_{x\in X} \inf_{y\in Y}f(x,y). \) The author presents a generalization of both these theorems: If one writes \(\text{med} (\alpha, \beta, \gamma)\) for the middle of the three numbers \(\alpha,\beta,\gamma\) and supposes for some number \(\tau\in \mathbb{R}\cup \{\pm\infty\}\) the conditions \[ \forall x_1,x_2\in X\;\exists x_0\in X\;\forall y\in Y:f(x_0,y) \geq\text{med} \bigl(f(x_1,y), f(x_2,y),\tau) \] \[ \forall y_1,y_2 \in Y\;\exists y_0\in Y\;\forall x\in X:f(x,y_0) \leq\text{med} \bigl(f(x,y_1), f(x,y_2),\tau) \] to be satisfied, then \(\inf_{y\in Y} \sup_{x\in X} f(x,y)= \sup_{x\in X} \inf_{y\in Y} f(x,y)\).
scientific article; zbMATH DE number 572401