On the semialgebraic Stone-Čech compactification of a semialgebraic set (Q2839941)

The authors study compactifications of a semi-algebraic set \(M \subseteq \mathbb{R}^n\). They show that there exists a universal compactification \(\hat{M}\), which they call the \textit{semi-algebraic Stone-Čech compactification}. The name is due to the analogy with the Stone-Čech compactification in topology. They give three different presentations of \(\hat{M}\), namely as the space of closed points of the prime spectrum of the ring of continuous semi-algebraic functions on \(M\), or as the space of closed points of the prime spectrum of the ring of bounded continuous semi-algebraic functions on \(M\), or as the projective limit of all semi-algebraic spaces that compactify \(M\). The space \(\hat{M}\) is rarely semi-algebraic. The authors study properties of the growth \(\hat{M} \setminus M\), in particular the number of connected components.NEWLINENEWLINEMany results in the paper are special cases of far more general properties of real closed rings. Every ring of continuous semi-algebraic functions is a real closed ring. The connections between the spectra of a real closed ring and a convex subring (e.g., the subring of bounded functions in a ring of continuous semi-algebraic functions) are well-known, see e.g. [\textit{N. Schwartz}, The basic theory of real closed spaces. Regensburger Math. Schr. 15, 257 p. (1987; Zbl 0634.14014), Chapter V.7; in: Proceedings of the Curaçao conference, Netherlands Antilles, June 26--30, 1995. Dordrecht: Kluwer Academic Publishers. 277--313 (1997; Zbl 0885.46024); Manuscr. Math. 102, No. 3, 347--381 (2000; Zbl 0966.13018); Math. Nachr. 283, No. 5, 758--774 (2010; Zbl 1196.13015)]. In particular the existence of the semi-algebraic Stone-Čech compactification has been known for a long time.NEWLINENEWLINEThe authors do not explain the connections of their results with the existing literature, even though this would lead to significant simplifications and a deeper understanding.