Some variants of remote sublocales

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 and study some of their properties.