A note on J-spaces and relatively connected subsets

Abstract

In this note we present an investigation of J-spaces and other types of spaces related to J-spaces. We characterise these spaces using the  notion of relatively connected subsets. For a completely regular J-space X, we give a description of the βX-extension of idempotent  elements of C(X) and characterise completely regular J-spaces using idempotent elements of some subrings of C(X). A brief study of relatively connected subsets which leads to the definition of C-normal spaces is given. We show that this new class of C-normal spaces  contains the class of normal spaces as a proper subclass. Michael proved that βX r X is relatively connected in βX if and only if X is a  completely regular J-space. We further establish conditions which characterise completely regular J-spaces via some closed subsets of βX.