On the Equivalence of Continuity and Hemicontinuity for Preference Preorders

Sufficient conditions are provided for a possibly incomplete preference pre-order on a topological space to be closed in the product space if and only if it has closed upper and lower contour sets. Notably, it is shown that the two properties are equivalent if the domain of the preorder is a Hausdorff (T2) topological space. The two concepts are therefore identical in the overwhelming majority of cases that are of interest to economists, even when completeness is not assumed.