Abstract
In this paper, we present a new simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. In particular, we show that, for completely regular spaces, a topology is useful, if and only if it is separable, and every isolated chain of open and closed sets is countable. As a specific application to optimization theory, we characterize the continuous representability of all continuous total preorders, which admit both a maximal and a minimal element.
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We gratefully acknowledge many helpful suggestions of two anonymous referees.
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Communicated by Juan-Enrique Martinez Legaz.
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This paper is dedicated to the memory of Professor Gerhard Herden, who passed away on January 30, 2019. He was a friend and an exceptionally clever mathematician. We are deeply indebted to him.
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Bosi, G., Zuanon, M. Topologies for the Continuous Representability of All Continuous Total Preorders. J Optim Theory Appl 188, 420–431 (2021). https://doi.org/10.1007/s10957-020-01790-y
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DOI: https://doi.org/10.1007/s10957-020-01790-y