Fernando Galván, José FranciscoGamboa Mutuberria, José Manuel2023-06-202023-06-2020121088-685010.1090/S0002-9947-2012-05428-6https://hdl.handle.net/20.500.14352/42296In 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.engjournal articlehttp://www.ams.org/journals/tran/2012-364-07/S0002-9947-2012-05428-6/S0002-9947-2012-05428-6.pdfhttp://www.ams.org/restricted access512.7Semialgebraic functionmaximal spectrumsemialgebraic compactificationsemialgebraic Stone–Čech compactificationremainderGeometria algebraica1201.01 Geometría Algebraica