On hereditary properties of extremally disconnected frames and normal frames

Abstract

Hereditary extremal disconnectedness of frames and its equivalent form of complete extremal disconnectedness are the topic of this paper. We study them in parallel with the corresponding normality properties of frames. Among several characterizations, we show that a frame is hereditarily normal [resp., hereditarily extremally disconnected] if and only if all its open and dense sublocales [resp., closed and dense sublocales] are normal [resp., extremally disconnected]. Even if spaces typically have more sublocales than subspaces, hereditary normality and hereditary extremal disconnectedness are shown to be conservative extensions of the classical properties. Furthermore, we provide such a general setting that permits us to treat several variants of the concepts under study in a unified way. Some of the presented results are new for the traditional topological spaces.