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Generalized Debreu’s Open Gap Lemma and Continuous Representability of Biorders

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Abstract

We prove a generalization of Debreu’s Open Gap Lemma. Given any subset of the real line, this lemma guarantees the existence of a strictly increasing real function such that all the gaps of the image of the subset are open. Now we extend it to the case of n subsets of the real line and study the existence of a strictly increasing real function such that all the gaps of the image of each set are open. This function does not exist in general so, we characterize the cases in which it exists. This generalization is not equivalent to Debreu’s lemma working on the union of the n subsets, neither on the intersection. Then we use it in order to obtain new results on continuous representations of biorders. We also generalize the concept of biorder.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Pública de Navarra, Campus Arrosadía, 31006, Pamplona, Spain

    Asier Estevan

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  1. Asier Estevan
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Correspondence to Asier Estevan.

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The authors acknowledge financial support from the Ministry of Science and Innovation of Spain under grant MTM2012-37894-C02-02.

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Estevan, A. Generalized Debreu’s Open Gap Lemma and Continuous Representability of Biorders. Order 33, 213–229 (2016). https://doi.org/10.1007/s11083-015-9360-1

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  • DOI: https://doi.org/10.1007/s11083-015-9360-1

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