On Isbell's Density Theorem for bitopological pointfree spaces I

TL;DR

This paper extends Isbell's Density Theorem to point-free bitopological spaces using $d$-frames, characterizing extremal epimorphisms and the existence of smallest dense sub-objects, with implications for functoriality.

Contribution

It provides the first analogue of Isbell's Density Theorem for $d$-frames in point-free bitopology, including characterization of extremal epimorphisms and the existence of smallest dense sub-objects.

Findings

Existence of smallest dense sub-objects in $d$-frames

Characterization of extremal epimorphisms in $d$-frames

Extension of Isbell's Density Theorem to point-free bitopological spaces

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.