A Note on the Continuity of Preference Relations defined on Discrete Sets

Jefferson Donizeti Pereira Bertolai, Fábio Barbieri, Mirelle Jayme and Nathan Machado

Revista Brasileira de Economia - RBE, 2024, vol. 78, issue 1

Abstract: This note discusses continuity of preference relations defined in discrete sets.Special interest is dedicated to the preference relation $\succeq$ defined in $X\equiv \cup_{n=1}^\infty\{1-1/n\}$ and such that, for each pair of points $x$ and $y$ in $X$, we have $x\succeq y$ if $|x-1/2|\geq|y-1/2|$.The relativity property of the open set concept emerges as a fundamental ingredient for this discussion and, therefore, its importance for Economic Theory is reinforced in a tranparent way.The continuity of such a relation is shown to be a particular case of a more general result: "Every correspondence with discrete domain is continuous''.

Date: 2024
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