Further remarks on totally ordered representable subsets of Euclidean space |
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| Authors: | Juan C. Candeal,Esteban Indurá in ,Ghanshyam B. Mehta |
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| Affiliation: | aDepartamento de Análisis Económico, Universidad de Zaragoza, Facultad de Ciencias Económicas y Empresariales, c/Gran Vía 2–4, E-50005 Zaragoza, Spain;bDepartamento de Matemática e Informática, Uniersidad Pública de Navarra, Campus Arrosadía s.n., E-31006 Pamplona, Spain;cDepartment of Economics, University of Queensland, Brisbane, Queensland 4072, Australia |
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| Abstract: | We introduce the property of ? -norm-boundedness on totally ordered subsets of Euclidean spaces. We show that when a closed subset X of the Euclidean space  n, endowed with a continuous total order ?, is ? -norm-bounded, the order topology and the induced Euclidean topology must coincide on X. This generalizes a recent result by Beardon, proved on connected totally ordered subsets of Euclidean space, because on totally ordered closed subsets of  n connectedness is a particular case of ? -norm-boundedness. We also analyze necessary and sufficient conditions for the coincidence of both topologies, and discuss some extension to the infinite-dimensional context. |
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| Keywords: | Ordered sets and order topologies Utility functions Euclidean space Topological vector spaces Normed spacesi |
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