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 $τ_{\mathcal A}$ 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 $(β_{\mathcal A}X, \bar{\mathcal A})$ 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 $β_{\mathcal A}X$. Next, we show that $β_{\mathcal A}X$ 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 $σ$-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 $(X, τ_{\mathcal A})$. Some questions raised in the literature are also answered.