On variants and generalizations of J-frames

Abstract

A J-frame is a pointfree topology counterpart of the concept of a J-space. In this paper, we present a study of some frames satisfying conditions that are stronger (and some that are weaker) than those defining J-frames. Conditions under which these frames coincide with J-frames are established - pointing out similarities and differences among these frames in the process. Moreover, JC-frames are introduced by mimicking the defining conditions of a J-frame, modulo replacing “compact” with “connected.” We show that all connected frames are JC-frames, not conversely. Any frame that can be written as a join of two disjoint connected sublocales is a JC-frame. Upon recalling that a frame with no points is a J-frame if and only if it is connected, we infer that all non-spatial J-frames are JC-frames.