Title: Further remarks on totally ordered representable subsets of Euclidean space
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 Rn, 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 Rn 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.
Publication Year: 1996
Publication Date: 1996-01-01
Language: en
Type: article
Indexed In: ['crossref']
Access and Citation
Cited By Count: 5
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot