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 \((X,\tau )\) 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 \(\tau ^{*}\) on \((X,\tau )\), the existence of a semicontinuous utility representation on the bitopological space \((X,\tau ,\tau ^{*})\) 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 \((X,\tau _1,\tau _2)\) 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.