Some variants of remote sublocales

TL;DR

This paper explores variants of remote sublocales in locale theory, establishing their relationships with Booleanization and introducing new localic maps that preserve these remote sublocales.

Contribution

It introduces and analyzes variants of remote sublocales, their connection to Booleanization, and defines new localic maps called f-remote preserving maps.

Findings

The coframe of sublocales equals all sublocales remote from Booleanization.

The supplement of Booleanization is the largest sublocale *remote from Booleanization.

Conditions are given for localic maps to preserve remote sublocales.

Abstract

We introduce and study some variants of remote sublocales, namely sublocales that are remote from dense sublocales and those that are *remote from dense sublocales. We show that the coframe of sublocales coincides with the collection of all sublocales remote from the Booleanization. Furthermore, the supplement of the Booleanization of any locale is the largest sublocale *remote from the Booleanization. We give conditions on localic maps such that their induced localic image and pre-image functions preserve sublocales that are remote (resp. *remote) from dense sublocales. We introduce new types of localic maps called f-remote preserving maps.