On Some Properties of Essential Darboux Rings of Real Functions Defined on Topological Spaces

Abstract

This paper deals with rings of real Darboux functions defined on some topological spaces. Results are given concerning the existence of essential, as well as prime Darboux rings. We prove that, under some assumptions connected with the domain $X$ of the functions, the equalities: $D(X)=S_{lf}(X), S(X)=\dim(\Re )$ hold, where $D(X)$ is a ${\cal D}$-number of $X$, $S(X)$ ($S_{lf}(X)$) denotes the Souslin (lf-Souslin) number of $X$ and $\dim(\Re )$ is a Goldie dimension of an arbitrary prime Darboux ring $\Re$.