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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Vydané v:Journal of mathematical economics Ročník 25; číslo 4; s. 381 - 390
Hlavní autori: Candeal, Juan C., Induráin, Esteban, Mehta, Ghanshyam B.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Amsterdam Elsevier B.V 1996
Elsevier
Elsevier Sequoia S.A
Edícia:Journal of Mathematical Economics
Predmet:
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
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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 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.
AbstractList 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.
The property of less than or equal to-norm-boundedness on totally ordered subsets of Euclidean space is introduced. It is shown that when a closed subset of X of the Euclidean space R superscript n, endowed with a continuous total order less than or equal to, less than or equal to-norm-bounded, the order topology and the induced Euclidean topology must coincide on X. This generalizes a recent result by Beardon (1994), proved on connected totally ordered subsets of Euclidean space, because on totally ordered closed subsets of R superscript n connectedness is a particular case of less than or equal to-norm-boundedness. Necessary and sufficient conditions for the coincidence of both topologies are analyzed, and some extension to the infinite-dimensional context is discussed.
Author Candeal, Juan C.
Mehta, Ghanshyam B.
Induráin, Esteban
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10.1016/0304-4068(93)90045-M
10.2307/2371274
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10.1007/BF01213622
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Issue 4
Keywords Ordered sets and order topologies
Euclidean space
C60i
Normed spacesi
Utility functions
Topological vector spaces
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Snippet 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...
The property of less than or equal to-norm-boundedness on totally ordered subsets of Euclidean space is introduced. It is shown that when a closed subset of X...
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SubjectTerms 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
Title Further remarks on totally ordered representable subsets of Euclidean space
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