Further remarks on totally ordered representable subsets of Euclidean space

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 R n, endowed with a continuous total order ≾, is ≾ -norm-bounded, the order topology and the induced Euclidean topology must coincide on X. This...

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Bibliographic Details
Published in:Journal of mathematical economics Vol. 25; no. 4; pp. 381 - 390
Main Authors: Candeal, Juan C., Induráin, Esteban, Mehta, Ghanshyam B.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 1996
Elsevier
Elsevier Sequoia S.A
Series:Journal of Mathematical Economics
Subjects:
Decision making
Economic models
Economic theory
Euclidean space
Mathematical analysis
Mathematical economics
Normed spacesi
Norms
Ordered sets and order topologies
Spatial dimension
Studies
Subsets
Topological vector spaces
Utility functions
Ordered sets and order topologies
Euclidean space
C60i
Normed spacesi
Utility functions
Topological vector spaces
ISSN:0304-4068, 1873-1538
Online Access:Get full text
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Summary: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 R 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 R 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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ISSN:0304-4068
1873-1538
DOI:10.1016/0304-4068(95)00734-2