Existence of an optimal choice on a compact set of alternatives

Conditions, chiefly reported earlier, are summarized and discussed for existence oi optimal, in different senses, options in choice from a fixed compact set. These conditions are formulated in the languages of binary preference relationships and of criterial spaces. Formal tools, some of which have been reported earlier, are analyzed which make it possible to establish relationships between assertions made in the case of an infinite set of options in different languages, “binary” and “criterial”, or in one language but for different optimality types. These tools of equivalent transition prove that these existence conditions decompose into two classes of equivalent theorems, no matter what language and optimality types are used.