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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Separating points from closed convex sets over ordered fields and a metric for $\tilde {R}^n$


Author: Robert O. Robson
Journal: Trans. Amer. Math. Soc. 326 (1991), 89-99
MSC: Primary 12J15; Secondary 12D15, 14P10, 14P99
DOI: https://doi.org/10.1090/S0002-9947-1991-1091232-X
MathSciNet review: 1091232
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Abstract: Let $R$ be an arbitrary ordered field, let $\bar R$ be a real closure, and let $\tilde R$ and ${\tilde R^n}$ denote the real spectra of $\bar R[X]$ and $\bar R[{X_1}, \ldots ,{X_n}]$. We prove that a closed convex subset in ${R^n}$ may be separated from a point not in it via a continuous "linear" functional taking values in $\tilde R$ and that there is a $\tilde R$-valued metric on ${\tilde R^n}$. The methods rely on the ultrafilter interpretation of points in ${\tilde R^n}$ and on the existence of suprema and infima of sets in $\tilde R$.


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Keywords: Convex set, real spectrum, ordered fields, real closed field, metric space
Article copyright: © Copyright 1991 American Mathematical Society