A Characterization Framework for Stable Sets and Their Variants

The theory of optimal choice sets offers a well-established solution framework in social choice and game theory. In social choice theory, decision-making is typically modeled as a maximization problem. However, when preferences are cyclic -- as can occur in economic processes -- the set of maximal elements may be empty, raising the key question of what should be considered a valid choice. To address this issue, several approaches -- collectively known as general solution theories -- have been proposed for constructing non-empty choice sets. Among the most prominent in the context of a finite set of alternatives are the Stable Set (also known as the Von Neumann-Morgenstern set) and its extensions, such as the Extended Stable Set, the socially stable set, and the $m$-, and $w$-stable sets. In this paper, we extend the classical concept of the stable set and its major variants - specifically, the extended stable set, the socially stable set, and the $m$- and $w$-stable sets - within the framework of irreflexive binary relations over infinite sets of alternatives. Additionally, we provide a topological characterization for the existence of such general solutions.