Order extensions,budget correspondences,and rational choice

This article is concerned with extensions of a continuous ordering R on a set X to a subset P(X) of the power set of X. The underlying topology will be the Hausdorff metric topology. We will see that continuous extensions of R do not require that P(X) contain every nonempty finite subset of X. Therefore, the analysis can be applied to consumer theory and inverse choice functions. In analogy to these functions budget correspondences are established which relate alternatives x with certain subsets of X, according to the extended ordering.