Further remarks on totally ordered representable subsets of Euclidean space - Candeal, Juan C.; Induráin, Esteban; Mehta, Ghanshyam B. (Journal of mathematical economics, 1996)
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.