Real closed spaces

Abstract

] on M. Then 0 can be reconstructed from se by certain types of ring extensions. The same construction can be done starting from any ring A and any constructible subset K of the real spectrum X(A) of A. In this way one obtains locally ringed spaces which are called affine real closed spaces (real closed since these spaces can be viewed as generalizing real closed fields). A real closed space is a locally ringed space which has an open cover by affine real closed spaces. In particular, to any sa space (M, (9) with an open affine cover M = \JM