Separating points from closed convex sets over ordered fields and a metric for 𝑅̃ⁿ
| Autor: | Robert O. Robson |
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| Rok vydání: | 1991 |
| Předmět: | |
| Zdroj: | Transactions of the American Mathematical Society. 326:89-99 |
| ISSN: | 1088-6850 0002-9947 |
| DOI: | 10.1090/s0002-9947-1991-1091232-x |
| Popis: | Let R R be an arbitrary ordered field, let R ¯ \bar R be a real closure, and let R ~ \tilde R and R ~ n {\tilde R^n} denote the real spectra of R ¯ [ X ] \bar R[X] and R ¯ [ X 1 , … , X n ] \bar R[{X_1}, \ldots ,{X_n}] . We prove that a closed convex subset in R n {R^n} may be separated from a point not in it via a continuous "linear" functional taking values in R ~ \tilde R and that there is a R ~ \tilde R -valued metric on R ~ n {\tilde R^n} . The methods rely on the ultrafilter interpretation of points in R ~ n {\tilde R^n} and on the existence of suprema and infima of sets in R ~ \tilde R . |
| Databáze: | OpenAIRE |
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