Fernando Galván, José Francisco and Gamboa, J. M.
(2012)
*On the semialgebraic Stone-Čech compactification of a semialgebraic set.*
Transactions of the American Mathematical Society
(364).
3479-3511 .
ISSN 1088-6850

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Official URL: http://www.ams.org/journals/tran/2012-364-07/S0002-9947-2012-05428-6/S0002-9947-2012-05428-6.pdf

## Abstract

In the same vein as the classical Stone–ˇCech compactification, we prove in this work that the maximal spectra of the rings of semialgebraic and bounded semialgebraic functions on a semialgebraic set M ⊂ Rn, which are homeomorphic topological spaces, provide the smallest Hausdorff compactification of M such that each bounded R-valued semialgebraic function on M extends continuously to it. Such compactification β∗sM, which can be characterized as the smallest compactification that dominates all semialgebraic compactifications of M, is called the semialgebraic Stone– ˇ Cech compactification of M, although it is very rarely a semialgebraic set. We are also interested in determining the main topological properties of the remainder ∂M = β∗sM \M 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 M. Moreover, ∂M is locally connected and its local compactness can be characterized just in terms of the topology of M.

Item Type: | Article |
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Uncontrolled Keywords: | Semialgebraic function, maximal spectrum, semialgebraic compactification, semialgebraic Stone–Čech compactification, remainder |

Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 16315 |

References: | 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. Nicolas Bourbaki, General topology. Chapters 1–4, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition. Hans Delfs and Manfred Knebusch, Separation, retractions and homotopy extension in semialgebraic spaces, Pacific J. Math. 114 (1984), no. 1, 47–71. 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 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 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 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 J.F. Fernando, J.M. Gamboa: On the spectra of rings of semialgebraic functions. Collect. Math., to appear (2012). 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 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 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. James R. Munkres, Topology: a first course, Prentice-Hall Inc., Englewood Cliffs, N.J., 1975. |

Deposited On: | 11 Sep 2012 08:59 |

Last Modified: | 02 Mar 2016 14:37 |

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