Continuous Preference Orderings Representable by Utility Functions

The paper surveys the conditions under which it is possible to represent a continuous preference ordering using utility functions. These contributions concern both the preference orderings and the spaces where they are defined. For any continuous preference ordering, we show the need for separability and the sufficiency of connectedness and separability, or second countability, of the space where it is defined. We emphasize the need for separability by showing that in any nonseparable metric space, there are continuous preference orderings without utility representation. However, by reinforcing connectedness, we show that countably boundedness of the preference ordering is a necessary and sufficient condition for the existence of
a (continuous) utility representation. Finally, we discuss the special case of strictly monotonic preferences.