On the Existence of Greatest Elements and Maximizers

We obtain several characterizations of the existence of greatest elements of a total preorder. The characterizations pertain to the existence of unconstrained greatest elements of a total preorder and to the existence of constrained greatest elements of a total preorder on every nonempty compact subset of its ground set. The necessary and sufficient conditions are purely topological and, in the case of constrained greatest elements, are formulated by making use of a preorder relation on the set of all topologies that can be defined on the ground set of the objective relation. Observing that every function into a totally ordered set can be naturally conceived as a total preorder, we then reformulate the mentioned characterizations in the more restrictive case of an objective function with a totally ordered codomain. The reformulations are expressed in terms of upper semi- and pseudo-continuity by showing a topological connection between the two notions of generalized continuity.