On the existence of greatest elements and maximizers
In this paper, several characterizations are obtained for the existence of greatest elements of a total preorder. Section 1 is introductory; and Section 2 provides some preliminary facts. In Section 3, necessary and sufficient conditions are given for a total preorder to have unconstrained greatest elements. The following important result is established there. Theorem 1. Let \(X\) be a set, \(B\) be a total preorder on \(X\), and \(\tau\) be the lower-order topology associated to \(B\). If \(Y\subseteq X\) and \(\mathcal{M}(B,Y)\ne \emptyset\), then \(Y\in \mathcal{K}(X,\tau)\). In Section 4, necessary and sufficient conditions are given for a function to admit unconstrained maximizers. A basic result established there is as below. Theorem 2. Let \(X\) be a set, \((T,\succeq)\) be a totally ordered set, and \(f:X\to T\) be given. The coarsest topology on \(X\) for which \(f\) is upper semi-continuous is compactly equivalent with the coarsest topology on \(X\) for which \(f\) is upper pseudo-continuous. Finally, Section 5 is devoted to some observations and examples; and Section 6, to certain conclusions to be derived. Further aspects occasioned by these developments are also being discussed.
scientific article; zbMATH DE number 7612879