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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Separating points from closed convex sets over ordered fields and a metric for $\tilde {R}^n$
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by Robert O. Robson PDF
Trans. Amer. Math. Soc. 326 (1991), 89-99 Request permission

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
  • © Copyright 1991 American Mathematical Society
  • 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