Abstract
We present different conditions for the existence of a pair of upper semicontinuous functions representing an interval order on a topological space without imposing any restrictive assumptions neither on the topological space nor on the representing functions. The particular case of second countable topological spaces, which is particularly interesting and frequent in economics, is carefully considered. Some final considerations concerning semiorders finish the paper.
Highlights
An interval order on a set X can be thought of as the simplest model of a binary relation on X whose asso ciated preference-indifference relation is not transitive
The existence of numerical representations of interval orders was first studied by Fishburn [1] [2] and by other authors
When the set X is endowed with a topology τ, it may be of interest to look for continuous or at least semicontinuous representations of an interval order on ( X,τ )
Summary
An interval order on a set X can be thought of as the simplest model of a binary relation on X whose asso ciated preference-indifference relation is not transitive. When the set X is endowed with a topology τ , it may be of interest to look for continuous or at least semicontinuous representations of an interval order on ( X ,τ ) Results in this direction were presented by Bosi. The existence of undominated maximal elements can be guaranteed by means of an approach of this kind (see, e.g., Alcantud et al [10]) This kind of semicontinuous representability of interval orders was first studied by Bridges [11] and by Bosi and Zuanon [12] [13]. We present different results concerning the representability of an interval order on a topological space ( X ,τ ) by means of a pair (u, v) of upper semicontinuous real-valued functions
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