×

On the semialgebraic Stone-Čech compactification of a semialgebraic set. (English) Zbl 1281.14046

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 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.
Many 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. [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.
The 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.

MSC:

14P10 Semialgebraic sets and related spaces
13J30 Real algebra
54C30 Real-valued functions in general topology
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors. · Zbl 0912.14023
[2] Nicolas Bourbaki, General topology. Chapters 1 – 4, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition. Nicolas Bourbaki, General topology. Chapters 5 – 10, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition.
[3] Hans Delfs and Manfred Knebusch, Separation, retractions and homotopy extension in semialgebraic spaces, Pacific J. Math. 114 (1984), no. 1, 47 – 71. · Zbl 0548.14008
[4] J.F. Fernando: On chains of prime ideals in rings of semialgebraic functions. Preprint RAAG (2010). http://www.mat.ucm.es/\( \sim \)josefer/pdfs/preprint/chains.pdf
[5] J.F. Fernando: On distinguished points of the remainder of the semialgebraic Stone-Čech compactification of a semialgebraic set. Preprint RAAG (2010). http://www.mat.ucm.es/\( \sim \)josefer/pdfs/preprint/remainder.pdf
[6] J.F. Fernando, J.M. Gamboa: On Łojasiewicz’s inequality and the Nullstellensatz for rings of semialgebraic functions. Preprint RAAG (2010). http://www.mat.ucm.es/\( \sim \)josefer/pdfs/preprint/null-loj.pdf · Zbl 1314.14106
[7] J.F. Fernando, J.M. Gamboa: On the Krull dimension of rings of semialgebraic functions. Preprint RAAG (2010). http://www.mat.ucm.es/\( \sim \)josefer/pdfs/preprint/dim.pdf · Zbl 1362.14059
[8] J.F. Fernando, J.M. Gamboa: On the spectra of rings of semialgebraic functions. Collect. Math., to appear (2012). · Zbl 1291.14085
[9] J.F. Fernando, J.M. Gamboa: On Banach-Stone type theorems in the semialgebraic setting. Preprint RAAG (2010). http://www.mat.ucm.es/\( \sim \)josefer/pdfs/preprint/homeo.pdf
[10] Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. · Zbl 0093.30001
[11] Giuseppe De Marco and Adalberto Orsatti, Commutative rings in which every prime ideal is contained in a unique maximal ideal, Proc. Amer. Math. Soc. 30 (1971), 459 – 466. · Zbl 0207.05001
[12] James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. · Zbl 0306.54001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.