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Transactions of the American Mathematical Society
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
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1091232-X
PII: S 0002-9947(1991)1091232-X
Keywords: Convex set, real spectrum, ordered fields, real closed field, metric space
Article copyright: © Copyright 1991 American Mathematical Society