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

TL;DR

This paper investigates the topological structure of the remainder in the semialgebraic Stone-Cech compactification of a semialgebraic set, distinguishing points based on local properties and analyzing associated ideal sets.

Contribution

It characterizes the points with metrizable neighborhoods in the remainder and studies the density and neighborhood systems of specific ideal-related point sets.

Findings

Points with metrizable neighborhoods are characterized by specific subsets.

Sets of free maximal ideals are dense in the remainder.

Points in these sets have countable neighborhood bases.

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 $\beta_s^* M$.