An introduction to locally semialgebraic spaces (Q1071833)

Semi-algebraic geometry deals with subsets \(M\subset R^ n\) (R an arbitrary real closed field) which are boolean combinations of finitely many sets of the form \(\{x\in R^ n| \quad P(x)\geq 0\}\) where \(P\in R[X_ 1,...,X_ n]\). To study the geometry of such sets the authors introduced the category of semi-algebraic spaces over the real closed field R. However, certain questions cannot be treated satisfactorily in this setting. A case in point is the classification of coverings. This necessitates the introduction of a larger category, the category of locally semi-algebraic spaces. This paper contains the definition, examples and a few properties of these spaces. A full account of the theory of locally semi-algebraic spaces may be found in the authors' book [''Locally semi-algebraic spaces'', Lect. Notes Math. 1173 (1985; Zbl 0582.14006)].