OPEN GAPS, METRIZATION AND UTILITY

It is shown that each of the Debreu Open Gap Theorem and the Debreu Co ntinuous Utility Representation Theorem can be used in order to prove the other. Furthermore, it is proved that the classical Alexandroff-Ur ysohn Metrization Theorem implies Debreu's Continuous Utility Represen tation Theorem and, thus, all known results on the existence of contin uous utility functions.