On the semialgebraic Stone–Čech compactification of a semialgebraic set

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 $M$.