On Isbell's Density Theorem for bitopological pointfree spaces I
M. Andrew Moshier, Imanol Mozo Carollo, Joanne Walters-Wayland
TL;DR
The paper extends Isbell's Density Theorem to point-free bitopology using the framework of ${ t dFrm}$, proving that every ${ t dFrm}$ has a smallest dense extremal epimorphism and hence a smallest dense sub-d-locale. Density is characterized by the reflection of the ${ m con}$ relation, and extremal epis require surjectivity on both components with appropriate closure of ${ m con}$ and ${ m tot}$; however, the image need not be Boolean, distinguishing the theory from frames. The authors develop a comprehensive sub-d-locale lattice, show how double pseudocomplements interact with density (often not forming sublocales), and introduce notions of corrigibility and dual subfitness to study functoriality, yielding a Skeletal-morphism–restricted endofunctor on ${ t dFrm}$. They also establish coreflection results for double negation frames and outline a program for extending these ideas to biframes in a sequel. The work lays groundwork for a robust theory of Isbell-type density in point-free bitopology and its categorical behavior, with potential implications for related structures in locale theory.
Abstract
This paper addresses dense sub-objects for point-free bitopology in terms of $d$-frames and provides several examples. We characterize extremal epimorphisms in $d$-frames and show that a smallest dense one always exists, establishing a proper analogue of Isbell's Density Theorem for $d$-frames. Further we explore certain questions about the functoriality of assigning the smallest dense sub-object to each pointfree bitopological space.