On the semialgebraic Stone-Čech compactification of a semialgebraic set

Authors:
José F. Fernando and J. M. Gamboa

Journal:
Trans. Amer. Math. Soc. **364** (2012), 3479-3511

MSC (2010):
Primary 14P10, 54C30; Secondary 12D15, 13E99

Published electronically:
February 29, 2012

MathSciNet review:
2901221

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Abstract: In the same vein as the classical Stone-Čech compactification, we prove in this work that the maximal spectra of the rings of semialgebraic and bounded semialgebraic functions on a semialgebraic set , which are homeomorphic topological spaces, provide the smallest Hausdorff compactification of such that each bounded -valued semialgebraic function on extends continuously to it. Such compactification , which can be characterized as the smallest compactification that dominates all semialgebraic compactifications of , is called the *semialgebraic Stone-Čech compactification of *, although it is very rarely a semialgebraic set. We are also interested in determining the main topological properties of the remainder and we prove that it has finitely many connected components and that this number equals the number of connected components of the remainder of a suitable semialgebraic compactification of . Moreover, is locally connected and its local compactness can be characterized just in terms of the topology of .

**[BCR]**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. MR**1659509****[Bo]**Nicolas Bourbaki,*General topology. Chapters 1–4*, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition. MR**979294****[DK]**Hans Delfs and Manfred Knebusch,*Separation, retractions and homotopy extension in semialgebraic spaces*, Pacific J. Math.**114**(1984), no. 1, 47–71. MR**755482****[Fe1]**J.F. Fernando: On chains of prime ideals in rings of semialgebraic functions.*Preprint RAAG*(2010).`http://www.mat.ucm.es/josefer/pdfs/preprint/chains.pdf`**[Fe2]**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/josefer/pdfs/preprint/remainder.pdf`**[FG1]**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/josefer/pdfs/preprint/null-loj.pdf`**[FG2]**J.F. Fernando, J.M. Gamboa: On the Krull dimension of rings of semialgebraic functions.*Preprint RAAG*(2010).`http://www.mat.ucm.es/josefer/pdfs/preprint/dim.pdf`**[FG3]**J.F. Fernando, J.M. Gamboa: On the spectra of rings of semialgebraic functions.*Collect. Math.*, to appear (2012).**[FG4]**J.F. Fernando, J.M. Gamboa: On Banach-Stone type theorems in the semialgebraic setting.*Preprint RAAG*(2010).`http://www.mat.ucm.es/josefer/pdfs/``preprint/homeo.pdf`**[GJ]**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. MR**0116199****[MO]**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. MR**0282962**, 10.1090/S0002-9939-1971-0282962-0**[Mu]**James R. Munkres,*Topology: a first course*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR**0464128**

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Additional Information

**José F. Fernando**

Affiliation:
Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Email:
josefer@mat.ucm.es

**J. M. Gamboa**

Affiliation:
Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Email:
jmgamboa@mat.ucm.es

DOI:
http://dx.doi.org/10.1090/S0002-9947-2012-05428-6

Keywords:
Semialgebraic function,
maximal spectrum,
semialgebraic compactification,
semialgebraic Stone–Čech compactification,
remainder.

Received by editor(s):
June 4, 2010

Published electronically:
February 29, 2012

Additional Notes:
Both authors were supported by Spanish GAAR MTM2011-22435, Proyecto Santander Complutense PR34/07-15813 and GAAR Grupos UCM 910444

Dedicated:
Dedicated to José María Montesinos on the occasion of his 65$^{th}$ birthday

Article copyright:
© Copyright 2012
American Mathematical Society