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Topologies for the Continuous Representability of All Continuous Total Preorders

Author

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  • Gianni Bosi

    (Università di Trieste)

  • Magalì Zuanon

    (Università degli Studi di Brescia)

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.

Suggested Citation

  • Gianni Bosi & Magalì Zuanon, 2021. "Topologies for the Continuous Representability of All Continuous Total Preorders," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 420-431, February.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:2:d:10.1007_s10957-020-01790-y
    DOI: 10.1007/s10957-020-01790-y
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    References listed on IDEAS

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    1. Herden, G., 1991. "Topological spaces for which every continuous total preorder can be represented by a continuous utility function," Mathematical Social Sciences, Elsevier, vol. 22(2), pages 123-136, October.
    2. Herden, G., 1989. "On the existence of utility functions," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 297-313, June.
    3. Herden, G., 1989. "On the existence of utility functions ii," Mathematical Social Sciences, Elsevier, vol. 18(2), pages 107-117, October.
    4. Campion, Maria J. & Candeal, Juan C. & Indurain, Esteban, 2006. "The existence of utility functions for weakly continuous preferences on a Banach space," Mathematical Social Sciences, Elsevier, vol. 51(2), pages 227-237, March.
    5. Bosi, Gianni & Herden, Gerhard, 2019. "The structure of useful topologies," Journal of Mathematical Economics, Elsevier, vol. 82(C), pages 69-73.
    6. Candeal, Juan C. & Herves, Carlos & Indurain, Esteban, 1998. "Some results on representation and extension of preferences," Journal of Mathematical Economics, Elsevier, vol. 29(1), pages 75-81, January.
    7. Bosi, Gianni & Herden, Gerhard, 2012. "Continuous multi-utility representations of preorders," Journal of Mathematical Economics, Elsevier, vol. 48(4), pages 212-218.
    8. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, October.
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    Cited by:

    1. Federico Quartieri, 2022. "On the Existence of Greatest Elements and Maximizers," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 375-389, November.
    2. Gianni Bosi & Roberto Daris & Gabriele Sbaiz, 2024. "New characterizations of completely useful topologies in mathematical utility theory," Papers 2402.18324, arXiv.org, revised May 2024.

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