On the semialgebraic Stone–Čech compactification of a semialgebraic set
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- by José F. Fernando and J. M. Gamboa PDF
- Trans. Amer. Math. Soc. 364 (2012), 3479-3511 Request permission
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 $M$.References
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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
- 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
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 364 (2012), 3479-3511
- MSC (2010): Primary 14P10, 54C30; Secondary 12D15, 13E99
- DOI: https://doi.org/10.1090/S0002-9947-2012-05428-6
- MathSciNet review: 2901221
Dedicated: Dedicated to José María Montesinos on the occasion of his 65$^{\text {th}}$ birthday