Continuity, completeness and the definition of weak preferences

This note explores the connections between continuity and completeness under alternative conceptions of preference relations. For non-trivial preorders, it shows that, unlike the standard definitions, the weak preference relation defined in Galaabaatar and Karni (2010) allows for incomplete preferences while maintaining all the continuity properties of complete preference relations. It also makes it possible to distinguish indifference between alternatives from non-comparability of alternatives. If the preference relations are complete, this definition agrees with the customary definitions.