A topological characterization of the existence of non-empty choice 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. In this paper, we characterize the existence of the most important solution theories of arbitrary binary relations over non-finite sets of alternatives. More precisely, we present a topological characterization of the Smith and Schwartz sets. We also generalize results of the above solution theories for asymmetric binary relations defined in finite sets as well as most of the known results concerning the (characterization of the) existence of maximal elements of binary relations on compact spaces.