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The structure of useful topologies

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  • Bosi, Gianni
  • Herden, Gerhard

Abstract

Let X be an arbitrary set. A topology t on X is said to be useful if every complete and continuous preorder on X is representable by a continuous real-valued order preserving function. It will be shown, in a first step, that there exists a natural one-to-one correspondence between continuous and complete preorders and complete separable systems on X. This result allows us to present a simple characterization of useful topologies t on X. According to such a characterization, a topology t on X is useful if and only if for every complete separable system E on (X,t) the topology tE generated by E and by all the sets X∖E¯ is second countable. Finally, we provide a simple proof of the fact that the countable weak separability condition (cwsc), which is closely related to the countable chain condition (ccc), is necessary for the usefulness of a topology.

Suggested Citation

  • Bosi, Gianni & Herden, Gerhard, 2019. "The structure of useful topologies," Journal of Mathematical Economics, Elsevier, vol. 82(C), pages 69-73.
    • Handle: RePEc:eee:mateco:v:82:y:2019:i:c:p:69-73
    DOI: 10.1016/j.jmateco.2019.02.006
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    Citations

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    Cited by:

    1. 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.
    2. Gianni Bosi & Magalì Zuanon, 2020. "Topologies for the continuous representability of every nontotal weakly continuous preorder," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(2), pages 369-378, October.
    3. M. Ali Khan & Metin Uyanik, 2020. "Binary Relations in Mathematical Economics: On the Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot," Papers 2007.01952, arXiv.org.
    4. 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.
    5. 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.

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