Continuity, Completeness and the Definition of Weak Preferences Edi Karni ∗ Johns Hopkins University May 30, 2011 Abstract This note explores the connections between continuity and com- pleteness under alternative conceptions of preference relations. For non-trivial preorders, it shows that, unlike the standard definitions, the weak preference relation defined in Galaabataar 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. Keywords: Preference relations, continuity, Archimedean axiom, mixture continuity, complete preorders, incomplete preferences JEL classification numbers: D01, D81 ∗ I am grateful to Juan Dubra, Tsogbadral Galaabataar, Zvi Safra, two anonymous referees and an associate editor for their useful comments and suggestions. 1