On the remainder of the semialgebraic Stone-Cěch compactification of a semialgebraic set

Abstract

In this work we analyze some topological properties of the remainder $\partial M:={\beta }_s^{\ast }M\setminus M$ of the semialgebraic Stone-Cěch compactification ${\beta }_s^{\ast }M$ 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 ${\beta }_s^{\ast }M$ that admit a metrizable neighborhood in ${\beta }_s^{\ast }M$ equals $M_{\mathrm{lc}}\cup ({\mathrm{Cl}}_{{\beta }_s^{\ast }M}({\overline{M}}_{\le 1})\setminus {\overline{M}}_{\le 1})$ where $M_{\mathrm{lc}}$ is the largest locally compact dense subset of $M$ and ${\overline{M}}_{\le 1}$ is the closure in $M$ of the set of $1$-dimensional points of $M$. In addition, we analyze the properties of the sets $\hat{\partial }M$ and $\tilde{\partial }M$ 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 $\partial M\setminus \hat{\partial }M$ and $\hat{\partial }M\setminus \tilde{\partial }M$ are also dense subsets of $\partial M$. It holds moreover that all the points of $\hat{\partial }M$ have countable systems of neighborhoods in ${\beta }_s^{\ast }M$.

15 pages. arXiv admin note: substantial text overlap with arXiv:1310.6291