Representations of Preference Orderings

Arrow-Hahn assume binary relation choose closed convergence closed subsets completes the proof continuous function continuous order isomorphism continuous total preorder continuous utility function contradiction countable set Debreu Debreu order separability defined Definition differentiable disjoint element equivalent euclidean exists a continuous exists a real-valued follows function f Gap Lemma Hausdorff metric Hence implies interval order irreflexive Jaffray Journal of Mathematical locally nonsatiated lower semicontinuous manifold mapping Mathematical Economics Mehta Nachbin's Neuefeind nonempty normally preordered Open Gap open sets open subset order dense subset order topology ordered set pair points preference relations preordered set preordered topological space Proposition rational numbers real number real-valued function real-valued order isomorphism representation second countable sequence strict partial order strict weak order strongly pseudotransitive subset of RN Suppose topological space topology of closed transitive upper semicontinuous vector whence ρίξ ди