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On the semialgebraic Stone-Čech compactification of a semialgebraic set

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Fernando Galván, José Francisco and Gamboa Mutuberria, José Manuel (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
Uncontrolled Keywords:Semialgebraic function, maximal spectrum, semialgebraic compactification, semialgebraic Stone–Čech compactification, remainder
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:16315
References:

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Deposited On:11 Sep 2012 08:59
Last Modified:07 Feb 2014 09:27

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