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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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 $ M\subset \mathbb{R}^n$, which are homeomorphic topological spaces, provide the smallest Hausdorff compactification of $ M$ such that each bounded $ \mathbb{R}$-valued semialgebraic function on $ M$ extends continuously to it. Such compactification $ \beta _s^*M$, which can be characterized as the smallest compactification that dominates all semialgebraic compactifications of $ M$, is called the semialgebraic Stone-Čech compactification of $ M$, although it is very rarely a semialgebraic set. We are also interested in determining the main topological properties of the remainder $ \partial M=\beta _s^*M\setminus 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, $ \partial M$ is locally connected and its local compactness can be characterized just in terms of the topology of $ M$.


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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
PII: S 0002-9947(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