On the frame of closed ideals and unifications of classes of spaces related to  <i>J</i> -spaces

Abstract In this paper, we explore the lattice of ideals of closed subsets of a topological space with an aim of utilizing these ideals to provide a unified approach to the study of J -spaces and related classes of topological spaces. Among other things, we show that the lattice of ideals of closed subsets of any topological space is a compact frame, and that this frame is connected if and only if the topological space at hand is connected. If two spaces are homeomorphic, the corresponding lattice of ideals of closed subsets are isomorphic as frames. Many results with a ring-theoretic flair are obtained. In particular, there is a sublattice of the lattice of ideals of C ( X ), defined by ideals of closed subsets of X , such that if C ( X ) is isomorphic to some ring C ( Y ), then so are the corresponding sublattices.