Numerical representation of choice functions

Numerical representations of choice functions allow the expression of a problem of choice as a problem of finding maxima of real-valued functions, which requires that less information be defined and which is easier to work with. In this paper, the existence of numerical representations of choice functions by imposing assumptions directly on the choice function is obtained. In particular, it is proved that (IIA) is a necessary and sufficient condition for a choice function to be representable, if the set of alternatives is countable, while the conjunction of (IIA) and a ‘continuity condition’ is sufficient to ensure it in separable topological spaces.