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A simple characterization of the existence of upper semicontinuous order-preserving functions

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  • Gianni Bosi

    (University of Trieste)

  • Laura Franzoi

    (University of Trieste)

Abstract

We introduce an upper semicontinuity condition concerning a not necessarily total preorder on a topological space, namely strong upper semicontinuity, and in this way we extend to the nontotal case the famous Rader’s theorem, which guarantees the existence of an upper semicontinuous order-preserving function for an upper semicontinuous total preorder on a second countable topological space. We show that Rader’s theorem is not generalizable if we adopt weaker upper semicontinuity conditions already introduced in the literature. We characterize the existence of an upper semicontinuous order-preserving function for all strongly upper semicontinuous preorders on a metrizable topological space.

Suggested Citation

  • Gianni Bosi & Laura Franzoi, 2023. "A simple characterization of the existence of upper semicontinuous order-preserving functions," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 11(2), pages 203-210, October.
    • Handle: RePEc:spr:etbull:v:11:y:2023:i:2:d:10.1007_s40505-023-00251-9
    DOI: 10.1007/s40505-023-00251-9
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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. Bosi, Gianni & Herden, Gerhard, 2016. "On continuous multi-utility representations of semi-closed and closed preorders," Mathematical Social Sciences, Elsevier, vol. 79(C), pages 20-29.
    3. Bosi, Gianni & Zuanon, Magalì, 2014. "Upper semicontinuous representations of interval orders," Mathematical Social Sciences, Elsevier, vol. 68(C), pages 60-63.
    4. Ghanshyam B. Mehta, 1997. "A remark on a utility representation theorem of Rader (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(2), pages 367-370.
    5. Trout Rader, 1963. "The Existence of a Utility Function to Represent Preferences," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 30(3), pages 229-232.
    6. Herden, Gerhard & Levin, Vladimir L., 2012. "Utility representation theorems for Debreu separable preorders," Journal of Mathematical Economics, Elsevier, vol. 48(3), pages 148-154.
    7. Athanasios Andrikopoulos, 2011. "Characterization of the existence of semicontinuous weak utilities for binary relations," Theory and Decision, Springer, vol. 70(1), pages 13-26, January.
    8. 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.
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    Cited by:

    1. 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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