Representations of Preference Orderings

a₁ Arrow-Hahn binary relation Birkhoff Cantor order choose closed convergence closed subsets compact completes the proof continuous function continuous isotone function continuous order isomorphism continuous real-valued order continuous total preorder continuous utility function contradiction countable set countable subset Debreu order separability define Definition element equivalent exists a continuous exists a real-valued follows function f Hausdorff Hence implies interval order isotone Jaffray lim inf locally nonsatiated lower semicontinuous mapping Mehta nonempty normally preordered open sets open subset order dense subset order topology ordered set pair preference relations preordered set preordered topological space rational numbers real number real-valued function real-valued order homomorphism real-valued order isomorphism representation second countable sense of Debreu sequence strict partial order strict weak order strictly increasing strictly isotone Suppose Theory topology of closed totally preordered set transitive upper semicontinuous vector whence