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We provide necessary and sufficient conditions for a correspondence takingvalues in a finite-dimensional Euclidean space to be open so as to revisit thepioneering work of Schmeidler (1969), Shafer (1974), Shafer-Sonnenschein (1975)and Bergstrom-Rader-Parks (1976) to answer several questions they and theirfollowers left open. We introduce the notion of separate convexity for acorrespondence and use it to relate to classical notions of continuity whilegiving salience to the notion of separateness as in the interplay of separatecontinuity and separate convexity of binary relations. As such, we provide aconsolidation of the convexity-continuity postulates from a broadinter-disciplinary perspective and comment on how the qualified notionsproposed here have implications of substantive interest for choice theory.

Separately Convex and Separately Continuous Preferences: On Results of Schmeidler, Shafer, and Bergstrom-Parks-Rader