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The Continuous Representation Property in Utility Theory

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Advances in Topology and Their Interdisciplinary Applications

Part of the book series: Industrial and Applied Mathematics ((INAMA))

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Abstract

A topological space \((X,\tau )\) satisfies the Continuous Representation Property (CRP) if every continuous total preorder defined on X can be represented by a continuous order-preserving real-valued function. The relevance of this property is discussed in the context of economics and social sciences. Certain characterizations of CRP are presented in terms of other familiar topological properties. In addition, two extensions of CRP, one regarding the semicontinuous case and the other involving an algebraic environment, are also discussed.

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Notes

  1. 1.

    Although there are other expressions in the literature to refer to this property, such as those of a useful topology, the continuous representability property,...etc.; following the suggestion of Professor G.B. Mehta, we have decided to call it the Continuous Representation Property as in [13].

  2. 2.

    In economics and decision sciences, it is very often to refer to a real-valued order-preserving function as a utility function. We will use indistinctly both terminologies.

  3. 3.

    A linear separable system on X is called in Bosi and Herden [3] a complete separable system.

  4. 4.

    As usual \(\omega _{1}\) stands for the first uncountable ordinal number. The long line L is the lexicographic product of \(\omega _{1}\) and [0, 1), both endowed with their natural orders. Alternatively, L can be defined in the following way: Between each ordinal \(\alpha \) and its successor \(\alpha +1\) put one copy of the real interval (0, 1). Then L is the space obtained in this way endowed with its natural order.

  5. 5.

    The semicontinuous representation property was introduced by Bosi and Herden in [2] under the name of a completely useful topology.

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Authors and Affiliations

  1. Departamento de Análisis Económico, Facultad de Economía y Empresa, Universidad de Zaragoza, Gran Vía 2. 50005, Zaragoza, Spain

    Juan C. Candeal

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  1. Juan C. Candeal
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Correspondence to Juan C. Candeal .

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  1. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Santanu Acharjee

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Candeal, J.C. (2023). The Continuous Representation Property in Utility Theory. In: Acharjee, S. (eds) Advances in Topology and Their Interdisciplinary Applications. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-99-0151-7_3

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