A characterization of the generalized optimal choice set through the optimization of generalized weak utilities

It often happens that a binary relation R defined on a topological space $$(X,\tau )$$ ( X , τ ) lacks a continuous utility representation; see, e.g., (Peleg, in Econometrica 38:93–96, 1970, Example 2.1). But under an appropriate choice of a second topology $$\tau ^{*}$$ τ ∗ on $$(X,\tau )$$ ( X , τ ) , the existence of a semicontinuous utility representation on the bitopological space $$(X,\tau ,\tau ^{*})$$ ( X , τ , τ ∗ ) can be ensured (see Remark 1 in the text). On the other hand, the traditional notion of weak utility representation as defined by Peleg (Econometrica 38:93–96, 1970) cannot be used to characterize the generalized optimal choice set, which requires binary relations that allow cycles. The main result in this paper states that for any generalized upper tc-semicontinuous, separable, pairwise spacious and consistent binary relation R defined on a bitopological space $$(X,\tau _1,\tau _2)$$ ( X , τ 1 , τ 2 ) and any subset D of X, there exists a utility function which characterizes the generalized optimal choice set of R in D in terms of the maxima of this function.