The ultrafilter theorem in real algebraic geometry
type: Item
instance of: scholarly article
title: The ultrafilter theorem in real algebraic geometry
zbMATH Open document ID: 0715.14048
DOI: 10.1216/RMJ-1989-19-3-611
published in: Rocky Mountain Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: This paper contains some variations on the theme of the ``ultrafilter theorem'' which gives a homeomorphism between the real spectrum of the coordinate ring of a real algebraic set V (with its constructible topology), and the Stone space of the boolean algebra of semialgebraic subsets of V. The first three sections fall under the rubric ``Stone duality'', as indicated by the author. He describes a general situation where prime filters of [subbasic\(| basic| constructible]\) [open\(| closed| void]\) subsets of a set Y coincide with points of Y (it is a pity that frequent errors concerning the notation for these nine families of subsets occur). This applies when Y is a subset of \(2^ A\) closed for the Tychonoff topology, for instance to the real spectrum of A which is considered in the fourth section. In this case, filters of open or closed basic subsets are related to subsets of the ring A, via abstract Stellensätze.
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 06E15
zbMATH DE Number: 4177287
zbMATH Keywords: Stone space of semialgebraic subsets
zbMATH Keywords: ultrafilter theorem
zbMATH Keywords: real spectrum of the coordinate ring of a real algebraic set
zbMATH Keywords: Stellensätze
author: Gregory W. Brumfiel
reviewed by: Michel Coste
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4177287