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Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders

Author

Listed:
  • Gianni Bosi

    (Department of Economics, Business, Mathematics and Statistics, University of Trieste, Via A. Valerio 4/1, 34127 Trieste, Italy)

  • Laura Franzoi

    (Department of Economics, Business, Mathematics and Statistics, University of Trieste, Via A. Valerio 4/1, 34127 Trieste, Italy)

  • Gabriele Sbaiz

    (Department of Economics, Business, Mathematics and Statistics, University of Trieste, Via A. Valerio 4/1, 34127 Trieste, Italy)

Abstract

We investigate properties of strongly useful topologies , i.e., topologies with respect to which every weakly continuous preorder admits a continuous order-preserving function. In particular, we prove that a topology is strongly useful provided that the topology generated by every family of separable systems is countable. Focusing on normal Hausdorff topologies, whose consideration is fully justified and not restrictive at all, we show that strongly useful topologies are hereditarily separable on closed sets, and we identify a simple condition under which the Lindelöf property holds.

Suggested Citation

  • Gianni Bosi & Laura Franzoi & Gabriele Sbaiz, 2023. "Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders," Mathematics, MDPI, vol. 11(20), pages 1-9, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4335-:d:1262324
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    References listed on IDEAS

    as
    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. Bosi, Gianni & Herden, Gerhard, 2019. "The structure of useful topologies," Journal of Mathematical Economics, Elsevier, vol. 82(C), pages 69-73.
    5. 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.
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