On the remainder of the semialgebraic Stone-C\v{e}ch compactification of a semialgebraic set

Abstract

In this work we analyze some topological properties of the remainder $\partial M:=\beta_s^* M\setminus M$ of the semialgebraic Stone-C\v{e}ch compactification $\beta_s^* M$ of a semialgebraic set $M\subset{\mathbb R}^m$ in order to `distinguish' its points from those of $M$. To that end we prove that the set of points of $\beta_s^* M$ that admit a metrizable neighborhood in $\beta_s^* M$ equals $M_{\rm lc}\cup( {\rm Cl}_{\beta_s^* M}(\overline{M}_{\leq1})\setminus\overline{M}_{\leq1})$ where $M_{\rm lc}$ is the largest locally compact dense subset of $M$ and $\overline{M}_{\leq1}$ is the closure in $M$ of the set of $1$-dimensional points of $M$. In addition, we analyze the properties of the sets $\widehat{\partial}M$ and $\widetilde{\partial}M$ of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder $\partial M$ and that the differences $\partial M\setminus\widehat{\partial}M$ and $\widehat{\partial} M\setminus\widetilde{\partial}M$ are also dense subsets of $\partial M$. It holds moreover that all the points of $\widehat{\partial}M$ have countable systems of neighborhoods in $\beta_s^* M$.