On the continuous analogue of the Szpilrajn Theorem I

On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

“…(P3) Determine necessary and sufficient conditions for a topology t on X to have the Dushnik-Miller property, which means that every weakly continuous preorder on X is the intersection of its continuous total refinements. Problem (P1) already has been discussed extensively in Herden and Pallack [32]. Therefore we concentrate in this paper solely on Problem (P2) and Problem (P3).…”

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

“…Example 3.2). A first still preliminary discussion of how to extend the concept of continuity for total preorders on X to arbitrary binary relations on X also can be found in Herden and Pallack [14,Section 2].…”

“…We have just to replace by R and ≺ by R S without any additional considerations. Theorem 3.1 implies, in particular, that the classical Debreu Representation Theorem can be extended to arbitrary binary relations (cf, Herden and Pallack [14,Theorem 2.15]). …”

“…This means that for every pair (x, y) ∈ R S there exists some z ∈ X such that xR S zR S y. A sketch of proof of this result already has been given in Herden and Pallack [14,Proposition 2.8]. A much more general and very detailed proof of this fact also has been given in Bosi and Herden [3,Proposition 2.3].…”

On the cardinality of indifference classes

“…Meanwhile the first author has obtained some partial results on the characterization of Hausdorff-topologies t on X that have the property that every non-empty closed subset of X is an indifference class of some continuous total preorder " " on (X, t) (cf. [10]). Clearly, these topologies must be completely regular.…”