Topological extensions of elementary measurable spaces

Abstract

An elementary space is a pair $(X,\mathcal{A})$, where $\mathcal{A}$ is an algebra of subsets of $X$. Every elementary space $(X,\mathcal{A})$ is naturally a topological space when equipped with the topology ${\tau }_{\mathcal{A}}$ generated by $\mathcal{A}$. Elementary mappings on elementary spaces are defined in a manner analogous to measurable mappings on measurable spaces. In addition, the notion of an {\it extension} of an elementary space is introduced. Let $M(X,\mathcal{A})$, denote the ring of all real-valued elementary functions on $(X,\mathcal{A})$. In this article, we study the ring $M(X,\mathcal{A})$ and extensions of an elementary space. We show that, for any elementary space $(X,\mathcal{A})$, there exists a unique compact elementary extension $({\beta }_{\mathcal{A}}X,\bar{\mathcal{A}})$ of $X$, analogous to the Stone-$\check{C}$ech compactification, which contains every other extension of $X$. This extension has the property that every elementary map from $X$ into any compact topological space extends continuously to ${\beta }_{\mathcal{A}}X$. Next, we show that ${\beta }_{\mathcal{A}}X$ contains a realcompact extension $({\upsilon }_{\mathcal{A}}X,{\bar{\mathcal{A}}}_R)$ of $X$, such that every elementary map from $X$ into any realcompact topological space extends continuously to ${\upsilon }_{\mathcal{A}}X$. Moreover, ${\upsilon }_{\mathcal{A}}X$ is the smallest realcompact extension of $X$ and $(X,\mathcal{A})$ is realcompact if and only if $X={\upsilon }_{\mathcal{A}}X$. Also, we see that, every metric space equipped with the $\sigma$-algebra of all Borel sets is realcompact. Finally, we observe that the maximal ring of quotients of $M(X,\mathcal{A})$ is the direct limit of locally constant (continuous) functions on dense open subsets of $(X,{\tau }_{\mathcal{A}})$. Some questions raised in the literature are also answered.