Summary
It is shown that each of the Debreu Open Gap Theorem and the Debreu Continuous Utility Representation Theorem can be used in order to prove the other. Furthermore, it is proved that the classical Alexandroff-Urysohn Metrization Theorem implies Debreu's Continuous Utility Representation Theorem and, thus, all known results on the existence of continuous utility functions.
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Herden, G., Mehta, G.B. Open gaps, metrization and utility. Econ Theory 7, 541–546 (1996). https://doi.org/10.1007/BF01213666
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DOI: https://doi.org/10.1007/BF01213666