A minimal regular ring extension of $C(X)$

M. Henriksen ¹   ; R. Raphael ²   ; R. G. Woods ³

¹ Department of Mathematics Harvey Mudd College Claremont, CA 91711, U.S.A.
² Department of Mathematics Concordia University Montreal, Québec Canada H4B 1R6
³ Department of Mathematics University of Manitoba Winnipeg, Manitoba Canada R3T 2N2

Fundamenta Mathematicae, Tome 172 (2002) no. 1, pp. 1-17

Résumé

Let $G(X)$ denote the smallest (von Neumann) regular ring of real-valued functions with domain $X$ that contains $C(X)$, the ring of continuous real-valued functions on a Tikhonov topological space $(X, \tau )$. We investigate when $G(X)$ coincides with the ring $C(X, \tau _\delta )$ of continuous real-valued functions on the space $(X, \tau _\delta )$, where $\tau _\delta $ is the smallest Tikhonov topology on $X$ for which $\tau \subseteq \tau _\delta $ and $C(X, \tau _\delta )$ is von Neumann regular. The compact and metric spaces for which $G(X) = C(X, \tau _\delta )$ are characterized. Necessary, and different sufficient, conditions for the equality to hold more generally are found.

Détail

DOI : 10.4064/fm172-1-1

Keywords: denote smallest von neumann regular ring real valued functions domain contains ring continuous real valued functions tikhonov topological space tau investigate coincides ring tau delta continuous real valued functions space tau delta where tau delta smallest tikhonov topology which tau subseteq tau delta tau delta von neumann regular compact metric spaces which tau delta characterized necessary different sufficient conditions equality generally found