A topological characterization of the existence of w-stable sets

The theory of optimal choice sets is a solution theory that has a long and well-established tradition in social choice and game theories. Some of important general solution concepts of choice problems when the set of best alternatives does not exist (this problem occurs when the preferences yielded by an economic process are cyclic) is the Stable Set (Von Neumann-Morgenstern set) and its variants (Generalized Stable set, Extended Stable set, m-Stable set and w-Stable set). The theory of w-stable sets solution is more realistic because: (1) It solves the existence problem of solution; (2) It expands the notions of maximal alternative set and (3) The concept of stability is defined in such a way as to prevent a chosen alternative from being dominated by another alternative and sets this stability within the solution. In this paper, we present a topological characterization of the existence of w-Stable sets solution of arbitrary binary relations over non-finite sets of alternatives.