Rim-Perfect Compactifications of Frames and Zero-Dimensionally Embedded Remainders

Abstract

We introduce zero-dimensionally embedded (ZDE) sublocales as those sublocales S with the property that the ambient frame has a basis, elements of which induce open sublocales whose frontiers miss S. This notion is stronger than the traditional zero-dimensionality of a sublocale. A compactification of a frame is perfect if its associated right adjoint preserves disjoint binary joins. Herein, the class of rim-perfect compactifications of frames is introduced, and we show that it contains all the perfect ones. Indeed, not every rim-perfect compactification is perfect, but compactifications with a ZDE remainder do not distinguish between rim-perfectness and perfectness. The Freudenthal compactification has a ZDE remainder. We show that a frame L is rim-compact if and only if L has a compactification with a ZDE remainder. Several results concerning perfect compactifications and ZDE remainders are provided.