The ultrafilter theorem in real algebraic geometry (Q752126)

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.